On the Relation of the Methods 



OF 



Just Perceptible Differences 
and Constant Stimuli 



By 
SAMUEL W. FERNBERGER 




A thesis presented to the Faculty of the Graduate 
School of the University of Pennsylvania in partial ful- 
filment of the requirements for the degree of Doctor 
of Philosophy. 



1912 



On the Relation of the Methods 



OF 



Just Perceptible Differences 
and Constant Stimuli 



By 
SAMUEL W. FERNBERGER 



A thesis presented to the Faculty of the Graduate School of the University of 
Pennsylvania in partial fulfilment of the requirements for the degree of Doctor 
of Philosophy. 

1912 







»2> 






Gift 
The Uoi-o^rsHy 

3 JUN 13 




TABLE OF CONTENTS 

PAGE. 

I. Introduction I & 5 

1 . Statement of the problem 1 

2. Differences between the two methods 1 

A. Formal 2 

B. Experimental 2 

3. Historical background of the problem 3 

II. Arrangement of the experiments 7-18 

1 . The subjects 7 

2. Form and adjustment of stimuli 7 

3. Experimental arrangement 10 

A. Elimination of space errors 11 

B. Controlling of the time error 12 

C. Order of presentation of stimuli 12 

D. Adjustment of the variable stimulus 13 

E. The judgments 15 

III. The method of constant stimuli 19-48 

1. Form of the results 19 

2. Constancy of conditions throughout the experiment 19 

A. Coefficient of divergence 19 

a. Method of calculation 22 

b. Results 23 

c. Discussion 24 

B. The psychometric functions 28 

a. Theory and method of calculation 29 

b. Results 38 

c. Discussion 41 

3. Method of constant stimuli as a measure of sensa- 

tivity 46 

A. Interval of uncertainty 46 

B. Point of subjective equality 47 

C. Time error 47 



iv INDEX 

IV. The method of just perceptible differences 49-81 

1 . Various forms of the method 49 

2. Analysis of the four fundamental differences .... 49 

3. Calculation of the probabilities of these differences 

from the results of the method of constant 

stimuli 53 

A. Theory and form of the calculations 55 

4. The observed and calculated results of these differ- 

ences J2> 

5. Form of the calculations of the observed values. . 73 
A. The probable error as a measure of accuracy . . 75 

6. Comparison of the results by the method of con- 

stant stimuli with the observed and calcu- 
lated results by the method of just perceptible 
differences 78 



I. INTRODUCTION 

Psychophysics has evolved a number of procedures by means of 
which the sensitivity of a subject can be determined. These 
procedures differ widely as to the experimental arrangement 
which they require and as to the calculation to which the results 
are subjected, but they have the common purpose of measuring 
the sensitivity of the subject. We may say, in general, that a 
psychophysical method is a prescription for the collecting of data 
and their evaluation in such a way that the result enables us to 
compare the sensitivity of different subjects, or of the same 
subject under different conditions or at different times. All these 
methods agree in this one point, that by them we undertake to 
give measures of sensitivity. The quantities which are used as 
the measures of sensitivity are widely different and, perhaps, not 
always directly comparable. One is confronted with the situation 
that the comparison of the sensitivity of different subjects by a 
given method yields perfectly satisfactory results, but that these 
results do not always agree with those obtained by other methods. 
One is then led to ask, what the relation of these different meas- 
ures of sensitivity may be. This problem is frequently stated in 
the form of the question of the relation of the method of con- 
stant stimuli and the method of just perceptible differences. 

It has been frequently pointed out that the methods of constant 
stimuli and just perceptible differences show variations of an 
experimental or of a mathematical nature in consequence of which 
their results are not comparable. Numerous investigators have 
noted this fact and have made different classifications of the 
methods on the basis of differences in their experimental pro- 
cedure or in the treatment of the results. In all of these classifi- 
cations the method of just perceptible differences is always the 
representative of one group and the method of constant stimuli 
is given as an example of another group [cf. Titchener, Experi- 
mental Psych., II, II, pp. 315-318]. At present these two methods 
have been developed to a more or less standard form, evidenced 



2 SAMUEL W. FERNBERGER 

by the fact that different authors describe them almost uniformly. 
The method of just perceptible differences is described by Wundt 
[Phys. Psych., 5th Ed., vol. I, pp. 470] ; G. E. Miiller [Die 
Gesichtspunkte nnd die Tatsachen der Psychophysischen Metho- 
dik, pp. 179] and Titchener [Experimental Psychology, vol. II, 
part I, pp. 55-69; part II, pp. 99-143]. A description of the 
method of constant stimuli is to be found in G. E. Miiller [Ge- 
sichtspunkte, pp. 35] and in Titchener [Exp. Psych., II, part I, pp. 
92-118; part II, pp. 248-318]. 

The solution of the problem of the relation of the psychophysi- 
cal methods certainly would have been much simpler, had it been 
possible to subject the same experimental data to the different 
calculations. The results of experiments made according to these 
descriptions cannot be treated as material for both methods. It 
was therefore necessary to divide the experiments into two 
groups, in one of which the data was taken by the method of 
just perceptible differences and in the other by the method of con- 
stant stimuli; and this brought with it the difficulty of deciding 
whether a given difference between the results was due to chance 
variations; to changes in the attitude of the subject, or to differ- 
ences in the methods themselves. The differences in the arrange- 
ment of the experiments by the two methods, indeed, are so 
great that one may suspect the existence of differences in the 
attitude of the subject. 

It has been argued that the influence of expectation, found 
in the method of just perceptible differences, where the subject 
has a knowledge of the stimuli, is a serious handicap. This 
influence is obviously absent in the method of constant stimuli 
since the experiments are arranged in such a way that the subject 
is given no clue whatsoever, as to the objective relation of the 
stimuli. 

In any study of the relation of these two methods, an effort 
must be made to eliminate all such influences and to have the data 
in such form that any variations between the results will be due to 
differences in the methods themselves and not to any of the influ- 
ences due to the experimental procedure. One means of elimi- 
nating the influence of expectation is to mingle the experiments 



INTRODUCTION 3 

by the two methods in such a way that the subject has no informa- 
tion as to the stimuli compared. The conditions in the two groups 
of experiments will be very nearly identical, if the results are 
taken simultaneously, but these conditions may undergo certain 
changes in the course of the experimentation. This belief cannot 
be disposed of offhand since the collection of the data neces- 
sarily requires a considerable time. One of the most important 
of these conditions is the psychophysical make-up of the subject. 
It is very likely that the psychophysical make-up itself does not 
remain constant since at least one factor, namely practice, changes. 
The difficulty, therefore, is twofold and requires an investigation 
of the changes due to variations that have an experimental basis, 
and also an investigation of the purely formal character of the 
methods. These two sides of the problem must not be confused. 
It may well be that the two methods are formally identical but 
in actual experimentation do not give the same results, since 
they are performed under different conditions. The suspicion 
that the conditions are not the same for the methods of just per- 
ceptible differences and constant stimuli is very strong, since the 
entire experimental arrangement is such as to produce different 
attitudes on the part of the subject. Thus it cannot be expected 
that the two methods will give the same results unless the experi- 
ment is arranged in such a way as to make the conditions directly 
comparable. Many experimenters have noted the fact that if 
results were taken by one of these methods and compared with the 
results of the other, that they do not agree. We may mention 
Meinong [Ueber die Bedeutiing dest Weberschen Gesetses. Zeits. 
f. Psych, und Phys. der Sinnesorgane, XI, 1896, pp. 244] ; 
Ebbinghaus [Psych., I, pp. 504] ; Merkel [Phil. Studien, IV, 
p. 543] and Boas [Pflilger's Arch., XXVIII, 1882, pp. 562]. 
The investigation of the formal character of the methods is 
the problem which Urban set for himself, and he devised for this 
purpose the notion of the probability of a judgment, which logi- 
cally led to the notion of the psychometric functions. [The 
Application of Statistical Methods to the Problems of Psycho- 
Physics, 1908, pp. 106.] Forming a judgment on the comparison 
of two stimuli is a chance event and there exists a certain proba- 



4 SAMUEL W. FERNBERGER 

bility with which a certain judgment will occur under certain 
conditions. If our experimental data are extended enough, the 
observed relative frequency of that judgment may be regarded as 
an empirical determination of its unknown probability. One of 
the conditions of a judgment is obviously the intensity of the 
stimuli. The psychometric functions give the probabilities of 
the different judgments as functions of the intensities of the 
comparison stimulus. Urban showed that the method of just 
perceptible differences could be analysed by means of these notions 
and that its final results could be stated in terms of the probabili- 
ties of the different judgments and the intensities of the stimuli. 
Empirical determinations of the probabilities of the extreme judg- 
ments, i. e. of the judgment greater and smaller, are the basis 
for the calculation by the method of constant stimuli, and it is 
therefore possible to test the formal character of the methods 
under discussion on one and the same set of results. Experience 
showed that both methods of treatment gave essentially the same 
results. 

-We are, therefore, confronted with the fact that the methods 
of constant stimuli and just perceptible differences give the same 
results if the calculations are made on the same material; but 
different results if the materials are different. From this we con- 
clude that the methods are formally identical but that the condi- 
tions, under which the experimental data must be gained, are 
materially different; that is, that the methods favor different 
attitudes of the subject. One must, therefore, devise an experi- 
mental procedure by which the two methods can be performed 
under as nearly identical conditions as possible, in order to study 
the agreement of the results from the two methods. 

The collection of a large amount of material enables one to 
study incidentally, an entirely different problem ; that of whether 
the conditions remain approximately the same throughout the 
experiment, or if they undergo certain changes. If the physical 
conditions of an experiment are kept as constant as possible, so 
that no variations in the conditions can be attributed to them, we 
must attribute any varying conditions to a change in the psycho- 
physical make-up of the subject. One factor, at least, making 



INTRODUCTION 5 

such a change extremely probable is the one due to the practice 
acquired by the subject during the performance of the 
experiments. 

Extended material, therefore, enables one to study the influence 
of progressive practice. If the experiments are made on two 
subjects, one of whom has a high degree of training in this kind 
of experimentation, while the other has none, one has the oppor- 
tunity to study the influence of practice in a very advanced stage 
and to compare it with the practice in the initial stage. 

These factors have led to the selection of the experimental 
arrangement for this study. It enables us to investigate the prob- 
lems of the formal and experimental relations of the methods 
of just perceptible differences and constant stimuli, and inciden- 
tally, the effect of progressive practice. 

My thanks are due to Prof. F. M. Urban for suggesting this 
problem to me and for acting as a subject. I also have to thank 
Prof. E. B. Twitmyer for revision of manuscript. 



II. ARRANGEMENT OF EXPERIMENTS 

This paper is based on the results of experiments in lifted 
weights on two subjects. The experimentation began January 3, 
1912, and was completed February 20, 1912. During this time 
records were taken almost every day and many times during the 
morning and afternoon. Subject I, Dr. F. M. Urban, was highly 
trained in the technique of lifted weights and was the same as the 
one designated as subject II in a former study by Urban [Statisti- 
cal Methods]. Subject II, the writer, had some experience in 
psychological experimentation but, at the beginning of this experi- 
ment, had no training in judging small differences of lifted 
weights. He was given one day's practice, in order to become 
acquainted with the experimental procedure and with the sensa- 
tions produced by weights that differ but slightly in intensity. 
From the second day his judgments were recorded. 

The weights used in this experiment were hollow brass cylin- 
ders, closed at one end. They were approximately 2.5 inches in 
diameter and 1 inch high and the wall was 0.0625 inches thick; 
and they were brought to any desired weight by filling them with 
shot and parafine. Although these weights had the same outward 
appearance to the subject, each weight could be recognized by the 
experimenter by means of small numbers stamped on them with 
a steel die. The set consisted of 15 weights; of which 7 were 
standard weights of 100 grams each. The comparison weights 
for the set used with the method of constant stimuli were 84, 88, 
92, 96, 104 and 108 grams. There were also two weights of 84 
grams and 97.44 grams which served for the preparation of the 
variable stimulus in the method of just perceptible differences. 

These cylinders had been made slightly lighter than the weight 
desired and a very delicate adjustment could be obtained by 
inserting shot and parafine until they were heavier than the proper 
intensity. The parafine was then carefully scraped out until the 
desired weight was obtained. An effort was made to use as little 
parafine as possible since it is susceptible to greater variation from 



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8 SAMUEL W. FERNBERGER 

atmospheric changes than the shot. It was not deemed necessary 
to determine the weights within a smaller value that 5 mgr. The 
weights were tested daily for the first week and then once a week 
throughout the experiment. Whenever a weight was found to 
vary more than 10 mgr., it was readjusted, but this was necessary 
only nine times during the experiment. During the first three 
weeks only one adjustment was necessary and no single weight 
had to be adjusted more than once. Table I contains the weights 
of the cylinders and also the amount of variation discovered in 
them, these variations being given in -f- or — mgr. from the 
correct weight. The first column contains the numbers stamped 
on the cylinders ; the next column gives the correct weight of each, 
and the succeeding columns contain the variations of the cylinders 
found on the date at the head of the column. A star indicates 
that a cylinder was found to vary more than 10 mgr. and that it 
was readjusted. 

In the choice of the materials which compose the weight, there 
are three essential points to be considered. In the first place, the 
weight must show very little variation from atmospheric changes ; 
secondly, there must be no distinguishing marks on them by means 
of which the subject can tell one from another. And thirdly, 
they must be the same in temperature and must arouse the same 
tactile sensations. Various types of weights have been suggested 
in the past. The weights used by Fullerton and Cattell were 
wooden boxes weighted to the proper intensity with shot and raw 
cotton. They were 6 cms. in diameter and 3 cms. high [Fullerton 
and Cattell, The Perception of Small Differences, 1892, pp. 118]. 
Jastrow [Amer. Journal of Psych., V, p. 245] used weights of 
the same type. Galton [Inquiries into Human Faculty, 1883, 
p. 373] used weights that were made by placing shot in a cart- 
ridge shell. Urban [Stat. Meth., p. 1] used the same brass cylin- 
ders that were employed in the present study. Brown [The 
Judgment of Difference, Univ. of California Pub., V, I, No. 1, 
1910] used cylindrical tin boxes 2.5 cms. high and 4.5 cms. in 
diameter, which were weighted with shot and parafine. 

In the choice of weights for our experiment, the wooden boxes 
must be at once thrown out of consideration, since they do not 



ARRAXGEMEXT OF EXPERIMENTS 9 

fulfil the first of our requirements. The variations in these 
weights are quite considerable as was shown by Urban [Stat. 
Meth., p. 173]. In this study Urban gives a table showing the- 
variations in his set of weights, which were similar to those used 
in this experiment and a set of Cattell weights that were adjusted 
but not used. The cartridge weights are open to the same criti- 
cism, although probably not to as high a degree as the wooden 
weights. An effort should be made to have the weights consist of 
as anhygroscopic materials as possible. With the weights used 
in this experiment, the brass cylinders themselves and the shot 
are practically anhygroscopic. The parafine was included to keep 
the shot stationary' and to simplify the adjustment and readjust- 
ment of the weights. 

The weights were as nearly alike as it was possible to turn them 
out, making it impossible for the subject to distinguish between 
them. The table that follows contains the height and diameter 
of each weisrht to 0.001 of an inch, and it can be seen that these 





Height in 


Diameter in 


Weight 


inches 


inches 


1 


0.996 


2.466 


2 


0.998 


2468 


3 


0.908 


2.470 


4 


0.995 


2.470 


5 


0.997 


2.468 


6 


0.997 


2.464 


7 


0.098 


2.464 


9 


0.990 


2.468 


12 


0.997 


2.468 


13 


0.999 


2.469 


14 


0.098 


2.467 


15 


0.997 


2.468 


19 


0.996 


2.467 


52 


0.997 


2.466 


Blank 


0.996 


2.468 



differences can be disregarded. Numbers were stamped on them 
so that they could be identified by the experimenter. The sur- 
faces were polished and lacquered, rendering them similar to the 
touch. The weights were kept under exactly the same conditions 
and furthermore, they were handled the same number of times, 
so that there was no perceptible difference in temperature. Thus 
the cylinders used in this experiment seem to conform to all the 
requirements of a set of weights and furthermore, have the ad- 
vantage of being easily adjustable. 



jo SAMUEL W. FERNBERGER 

The weights were placed at regular intervals around the circum- 
ference of a circular table with a revolving top, which was 75.5 
cms. in diameter and was raised 68.0 cms. from the floor. The 
top was covered with a layer of prepared cork, which deadened 
the sound of the weights when replaced on the table. The posi- 
tion of each weight was indicated by a small number on the table. 
The standard and comparison weights were placed alternately — 
the standard weights at the odd, and the comparison weights at 
the even numbers on the table. 

The subject was seated in a comfortable position with his right 
arm supported by a table in such a way that the hand, from the 
wrist down, hung over the edge. An effort was made to have the 
edge of the supporting table strike approximately the same posi- 
tion of the forearm of the subject. The turn top table was then 
brought into such a position that, with merely a downward move- 
ment of the wrist, the hand would grasp one of the weights. 

The cylinders were lifted with the right hand ; most of the 
weight being sustained by the thumb, second and third fingers, and 
the first and little fingers resting on the edge. The movements 
of the hand were regulated by the beats of a metronome, which 
was adjusted to 92 beats per minute, while every fourth beat was 
accentuated by the automatic stroke of a bell. These hand move- 
ments were regulated in the following manner. At the start of 
each trial, the hand of the subject was raised at the wrist, with 
the forearm remaining on the table. At the stroke of the bell, 
the hand was dropped and the weight, which had been brought 
directly underneath by turning the table, was grasped. At the 
second stroke of the metronome the weight was lifted, and at 
the next stroke, it was replaced on the table. Finally at the 
fourth stroke, the empty hand was lifted, returning to its original 
position. Between the third stroke of the metronome and the 
bell following, the experimenter turned the table so that the next 
weight to be lifted, was directly under the hand of the subject, 
and everything was ready for the next lifting. In a very short 
time these wrist movements became quite automatic. The weights 
were lifted from 2 to 4 cms. and an effort was made to have the 
height of lifting constant for each subject. Due to the control 



ARRANGEMENT OF EXPERIMENTS n 

of the metronome, each weight was in the air approximately 
the same length of time. 

A screen was placed between the subject and the table so that 
it was impossible for the subject to see the weights; his hand 
passing through a slit in this screen. Furthermore both subjects 
voluntarily closed their eyes while making judgments, as they 
believed that they could make their judgments with more accuracy 
in that way. 

Previous investigators have not made a very clear analysis of 
the space error. They have sought to avoid it or eliminate its 
influence rather than explain it. The obvious method of elimina- 
ting the space error and the one that has been most frequently 
used, is to perform the experiment twice, in both of the spatial 
relations; that is, with the standard weight to the left and right 
of the comparison weight. Then the error of the one spatial 
relation counterbalances the error of the other, which is in 
the opposite direction. In the present study the space error was 
avoided in a simpler manner; since by means of the revolving top 
of the table, all side movements of the hand were eliminated. The 
table was turned so that the weight to be lifted was brought 
directly under the hand of the subject. Thus the only movements 
necessary were in one direction merely — directly downward. 
Care was taken that the subject did not reach to one side or the 
other in grasping a weight, since in this case a space error would 
have occurred. An effort has been made before to avoid the space 
error by experimental technique rather than eliminate it by repeat- 
ing the experiment in the two spatial relation. [L. Steffens, 
Zeitschrift f. Psych, und Physiol, der Sinnesorgane, XXIII, 
1900, pp. 279, and J. F robes, Zeit. f. Psych, und Physiol, der 
Sinnesorgane, XXXVI, 1904, pp. 234]. In these studies the 
weights were placed on a board which was pushed along under the 
hand of the subject, so that the weights in turn were directly 
underneath. This procedure, although it eliminated the space 
error, had the disadvantage that when a series had been taken, the 
board had to be replaced in its original position before a second 
series could be begun. With our experimental arrangement, how- 
ever, any number of series could be run off in succession. Besides 



12 



SAMUEL W. FERNBERGER 



this, the table, which revolves very easily, can be moved with 
greater regularity and accuracy than could ever hoped to be ob- 
tained with a sliding board. 

The time error was present in our experiments and no effort 
was made to either avoid or eliminate it. The standard stimulus 
was always lifted first and the comparison stimulus second. 
The metronome controlled all hand movements and kept them 
regular so that the time error was constant throughout the experi- 
ment. As the investigation of the sensitivity of the subjects was 
not of primary interest, and as a constant time error should not 
effect the relationship of the results obtained by the two methods 
under discussion, no attempt was made to avoid this error. 

The comparison weights were placed about the table in a care- 
fully arranged order. This order was changed four times dur- 
ing the experiment, partly to eliminate the influence of the 
particular arrangements, partly to counteract the influence of the 
knowledge about the arrangement used, which the subjects might 
have acquired. Table II gives these orders. The first column 



Table No. 


1/3/12 


1/10/12 


1/22/ 1 2 


2/18/12 


i and 2 


96 


84 


84 


88 


3 and 4 


104 


104 


104 


C 


5 and 6 


108 





C 


104 


7 and 8 


84 


C 


88 


96 


9 and 10 


02 


92 


92 


84 


11 and 12 


C 


108 


108 


92 


13 and 14 


88 


88 


96 


108 



Table II 



gives the table numbers in pairs : 1 being the first standard 
stimulus, 2 the first comparison stimulus, 3 the second standard 
stimulus and so forth. The other columns give the comparison 
weights at the even numbers of that pair. The C indicates the 
comparison weight of the method of just perceptible differences 
series. Each column is under the date at which the order was 
adopted. In every order, no matter how carefully planned, there 



ARRANGEMENT OF EXPERIMENTS 13 

are certain landmarks by means of which the subject can tell in 
what part of the series he is. Such a landmark is seen in the 
first series where the two heaviest weights (104 and 108 gms.) 
come together. Another occurs in the second order where the 
two lightest comparison weights [84 and 88 gms.] came together. 
Both subjects acted as experimenters and so necessarily became 
acquainted with the order in which the weights were presented. 
Furthermore, the experiments were conducted daily and so the 
subjects became acquainted with the orders more rapidly than if 
there had been a longer interval between experimentation. 

One complete revolution of the table involved the passing of 
seven judgments : six on invariable comparison weights for the 
method of constant stimuli, and one on a variable comparison 
weight for the method of just perceptible differences. 

This seventh comparison weight, indicated by C in table II, 
was adjusted for the method of just perceptible differences in 
the following manner. At each revolution of the table, a judg- 
ment was passed between it and a standard weight in the same 
manner as the other pairs. After a judgment had been taken on 
each of the seven pairs, steel bearings of a given number were 
placed in cylinder C and another complete revolution of the 
table was made. Then the same number of bearings were placed 
in cylinder C. This was continued until C weighed over 108 gms., 
the weight of the heaviest stimulus used in the method of constant 
stimuli. The bearings which weighed 0.42 gms., were of a sur- 
prising uniformity in weight. All the bearings were weighed 
and they did not show any variations within 5 mgr., which was 
the limit of exactitude in our weighing. We at first intended 
to use shot for the purpose of weighting our variable stimulus, 
but found that the differences among them were quite consider- 
able. Cotton wool was placed in the bottom of the cylinder so 
that the total weight was 84.80 gms. The cotton wool was 
placed in the cylinder to keep the bearings from moving about 
and thus by the noise, indicating to the subject which cylinder 
he was lifting. The bearings made a noise only three or four 
times during the lifting, and each time the judgment was thrown 
out and another taken. 



i 4 SAMUEL W. FERNBERGER 

Ten series were taken by means of this method. In series 
II the variation was two bearings for each revolution of the 
table. For series III the variation was three bearings, and so on 
until series X, when a variation of ten bearings was used. In 
an effort to test the "carefully graded approach" of the central 
intensities of the comparison stimulus, which is considered by 
some psychologists to be the keynote of the method of just per- 
ceptible differences, another series was planned [Titchener, Ex- 
perimental Psychology, II, II, p. 103]. In this series the two 
extreme variations were seven bearings; the next two variations 
toward the central values were six bearings; then in order five, 
four, three, two, and the five central values varied only one bear- 
ing. This is designated as series I. It was not deemed profitable 
to perform a series with the variation of only one bearing, as this 
would have necessitated 54 revolutions of the table to complete 
each series. 

Ten determinations of each series were taken; of these five 
were ascending and five descending. In the ascending series the 
proper number of bearings were placed in the cylinder, after each 
revolution of the table; while in the descending series, the total 
number of bearings were placed in the cylinder at the start of the 
experiment and after each revolution of the table, the proper 
number were removed. It is obvious that these series varied in 
length ; the series X required only seven revolutions of the table 
while series II required 29 revolutions. As a matter of technique, 
two weights were used for this comparison stimulus. The first 
with the cotton wool weighed 84.80 gms. and the second with the 
cotton wool, 98.24 gms. In an ascending series, the 84.80 gm. 
cylinder was first judged empty, then successive judgments were 
taken until it weighed equal to or just heavier than 98.24 gms. 
Then the other cylinder was substituted and the lifting continued. 
The opposite procedure was used in a descending series. The use 
of only one cylinder would have necessitated the handling of 
twice as many bearings as were used. 

The series were not taken in any regular order but entirely at 
haphazard; the determining factor in the choosing of a series 
being the amount of time at our disposal. If we had sufficient 



ARRANGEMENT OF EXPERIMENTS 15 

time, a long series of short steps was chosen ; while if our time was 
short, a short series of long steps was used. A different starting 
point was chosen for each successive revolution of the table. This 
was done so that, even though a subject knew an order fairly 
well, he would not be able to tell at what part of that order he 
had started. It was furthermore deemed advisable, on each revo- 
lution of the table, to allow the subject to make two judgments 
that were not recorded. This was done so that the movements 
of the wrist might become as automatic as possible and also to 
give the subject an opportunity to concentrate his attention. 

At the beginning of experimentation each day, the subjects 
made one complete revolution of the table grasping the weights 
as strongly as possible and lifting them high and vigorously. 
This was done for a double purpose : it assured the experimenter 
that the weights were in the correct position on the table in rela- 
tion to the hand of the subject. At the beginning of experimenta- 
tion, the weights give the impression of great lightness which is 
lost as the lifting process proceeds. This process can be hastened 
by the sort of lifting just described, and after such a warming up, 
both subjects made introspections that they had little trouble 
with the absolute impression. 

After each five to seven revolutions of the table, the subject 
was allowed to rest, until he was willing to resume experi- 
mentation. If a subject declared that he felt fatigued or unfit 
he was not asked to experiment. 

In the manner just described, results by the method of just 
perceptible differences were taken simultaneously, and therefore 
under as nearly indentical conditions as possible, with results by 
means of the method of constant stimuli. The former are the 
results obtained with the weight C and a standard weight, and 
the latter are the judgments on the six other pairs of weights. 

Immediately after each comparison weight had been replaced 
on the table, a judgment was given in terms of the comparison 
weight. These judgments were given verbally and by saying 
one word, three terms being used: heavier, equal and lighter. 
A heavier judgment signified that the comparison weight was 
subjectively heavier than the standard weight just preceding it. 



i6 



SAMUEL W. FERNBERGER 



A lighter judgment signified that the comparison weight was 
subjectively lighter than the standard weight just preceding it. 
The equality judgment was more complex as it not only included 
cases of actual subjective equality between the standard and 
comparison weights, but also all those cases where it was impos- 
sible for the subject to give either a lighter or a heavier judgment, 
usually termed doubtful cases. The cases of absolute subjective 
equality were much more frequent with subject I than with 
subject II. 

The results were recorded by the experimenter on printed 
blanks of the following form : 



96 


h 


e 


e 


104 


h 


h 


h 


108 


h 


h 


h 


84 


1 


1 


1 


92 


1 


e 


1 


c 


1 



1 
6 


1 

12 


88 


1 


1 


1 



The first column of these blanks gives the comparison weights, 
C being the variable weight for the method of just perceptible 
differences. The succeeding columns give the judgments for a 
complete revolution of the table, H signifying a heavier; E an 
equal, and L a lighter judgment. The small numbers (o, 6, 12, 
etc.) indicate the number of bearings in the comparison weight 
C during that revolution of the table. The above chart is a 
portion of the record of a series VI, as the weight C is regularly 
varied by six bearings. 

From Fechner down, there has been a great deal of discussion 
about the choice of judgments; Fechner himself, objecting to the 
use of the equality or doubtful judgments [Elemente der Psycho- 
physik, I, 72, 94. Revision der Hauptpunkte der Psychophysik, 
67]. He suggested that the equal and doubtful judgments be 
divided in half and that one half be added to the right and the 
other half to the wrong cases. This procedure was followed by 



ARRANGEMENT OF EXPERIMENTS 17 

Fullerton and Cattell [Perception of Small Differences, pp. 59]. 
Merkel suggested that the equal judgments be not only excluded 
from the calculations but also from the records [Philosophische 
Studien, IV, 131] and this method was followed out by Kraepelin 
in his experimental procedure [Philosophische Studien, VI, pp. 
496]. Jastrow [Amer. Jour, of Psych., I, 282] and Higier 
[Phil. Stud., VII, 247] contended that there should be no equal 
or doubtful judgments and that when the subject does not know 
whether the judgment should be greater or less, he should guess. 
Sanford [Course, pp. 357] follows this same procedure. Brown 
[Judgment of Difference, 1910] is the latest adherent of the 
exclusion of the equality or doubtful judgments. He even asserts 
that if the subject is forced to give a judgment of either greater 
or less, that he can do so. Brown apparently fails to notice that 
this places his uncertain judgments in exactly the same category 
as those of Jastrow where the subject is forced to guess. Nor 
should the numerical results of these variations differ widely 
from those of Fechner, because the laws of chance would give 
approximately an equal division of this class of judgments be- 
tween the two other classes. The only difference is that Fechner 
makes this division frankly while the others hide it under the 
technicalities of experimental procedure. 

Urban [Application of Statistical Methods, pp. 5] uses a very 
complicated system of judgments. He uses the classes of greater, 
equal and lighter. The degree of confidence with which the 
judgment is given is designated by the subject by the numerals 
1, 2 or 3. Besides these he uses two guess classes, "Guess- 
heavier and guess-lighter". Later in the paper [Stat. Meth., pp. 
14] he states that this system is too complicated and that the 
guess judgments are not advisable as they afford a loophole for 
the subject who does not wish to commit himself. Among the 
men who favor the use of the equality judgment we find Ebbing- 
haus, Wundt and Muller. 

Titchener [Experimental Psychology, vol. II, part II, pp. 290] 
points out the original reasons why the equal judgments were 
first excluded. He shows that it was because three classes pre- 
sented difficulties in mathematical treatment that were at that 



i8 SAMUEL W. FERNBERGER 

time unsurmountable. But these difficulties have since been 
solved and a further exclusion of judgments of this class 
for that reason, he calls unscientific. Titchener further points 
out that we have the evidence of introspection of trustworthy- 
observers that equal and doubtful judgments do actually 
occur. "If we are dealing with mind as mind presents itself to us 
for examination, we cannot ignore these judgments." This seems 
to be an unanswerable criticism. Brown, in his paper, gives a case 
of subjective equality. His subject stated [Judgment of Differ- 
ence, pp. 30] "those two are exactly the same". This may have 
been only one case in 4000, as Brown states, but still it must be 
accounted for. The subjects in our experiment both reported 
actual subjective equality; subject I gave this introspection quite 
frequently. The criticism of Brown [pp. 28] that the classes 
should be mutually exclusive is just as applicable to three classes 
as to two. His criticism that [pp. 32], for the equality judg- 
ment, the subject must maintain a mental standard of equality is 
not very impressive. Undoubtedly the subject must maintain 
such a standard, but he must also maintain, in the same sense, 
a standard for heaviness and lightness. 

The classes of judgments chosen for this experiment fulfil all 
the requirements of the case. They are susceptible to mathe- 
matical treatment and they are not so complicated that the sub- 
ject has any difficulty in giving them. The subject is not put 
to the strain of forcing a guess judgment. Lastly and of pri- 
mary importance they fit the facts of actual experience. 



III. METHOD OF CONSTANT STIMULI 

The record sheets enable us to find the relative frequencies 
of the heavier, equal and lighter judgments for all of the six 
comparison weights. In order to study the effect of practice, 
these observed frequencies were divided into groups of ioo in the 
order in which they were taken. These results are given in Table 
III for subject I and in Table IV for subject II. The numbers 
in the first column give the groups of ioo judgments in the order 
in which they were taken. Three columns are given to each com- 
parison weight in which appear the lighter, equal and heavier 
frequencies of the judgments on that weight. The numbers in 
the same columns show, on the whole, a rather close agreement. 
but that they are by no means identical. For this reason it is not 
possible to say offhand whether a constant effect of practice has 
taken place or whether the conditions remain the same. This 
decision may be affected in two ways ; by the determination of 
the coefficient of divergence for the probabilities for the different 
judgments, and by the determination of the constants for the 
psychometric functions. 

The data in tables III and IV are results of repeated observa- 
tions of certain unknown probabilities and the question arises 
whether the conditions, which determine these probabilities, 
remain the same or undergo variations. The coefficient of diver- 
gence [Lexis and Dormoy] enables us to make this decision 
systematically. • If the coefficient gives a value close to unity we 
have a normal dispersion, and may know that the conditions have 
been approximately constant during the experiment. If the 
coefficient of divergence is considerably smaller than unity we 
obtain an under normal dispersion and we conclude that there is 
a law or rule which tends to produce always the same value of the 
relative frequencies. If the value comes out considerably greater 
than one, we speak of a more than normal or over normal disper- 
. sion which indicates that the conditions during the experiment 
have varied. We may illustrate these three kinds of dispersion by 



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22 SAMUEL W. FERNBERGER 

considering the drawing from an urn of black and white balls 
of a given relative frequency. Several urns are prepared and n 
numbers of drawings are made from each urn, and each ball is 
replaced after its colour has been noted. Under such conditions 
if the number of drawings is great enough, we will obtain a 
coefficient of divergence approximating unity, or in other words, 
a normal dispersion. An over normal dispersion will be obtained 
when the urns, before the drawings are made, contain different 
relative frequencies of black and white balls. Again each ball 
will be replaced in the urn after its colour has been noted and the 
same number of drawings will be made from each urn. But the 
varying frequencies in the urns represent changed conditions so 
that our results will give a coefficient of divergence greater than 
unity. If we can by some kind of device eliminate certain 
cases we will obtain an undernormal dispersion. For example, 
if we decide that every time three white or black balls are drawn 
in succession we shall record the third ball as having the opposite 
colour, we would obtain a coefficient of divergence less than unity. 
From this we conclude that in an under normal dispersion the re- 
sults have been tampered with in some way. We must calculate 
the coefficient of divergence for the probabilities of each judgment 
for every intensity of the comparison stimulus that we used. The 
formula by which the coefficient of divergence is caluclated is 



o \ s * v * 

V - \(„— 1) p(\—p) 

in which s is the number of observations in each series; 2v 2 is 
the sum of the deviations of the relative frequencies from their 
average, n is the number of series and p the average of the rela- 
tive frequencies, which is the most probable determination of the 
probability of the judgment. The quantity (i < — p) will then be 
the probability that this judgment will not occur. Table V is a 
double table that gives the coefficients of divergence for both 
subjects I and II. In the first column are found the intensities of 
the comparison weights. The next three columns give the co- 
efficients of divergence for the heavier, lighter and equal judg- 
ments for subject I. In the vertical column, headed Average, is 
given the average of the coefficients of all three judgments for 





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24 SAMUEL W. FERNBERGER 

each comparison weight ; and in the horizontal column we find the 
averages of the vertical columns. The last value of the hori- 
zontal averages is the total average of all the coefficients of 
divergence. The second half of the table shows a similar dis- 
tribution for subject II. 

In the case of subject I, the total average, 1.154, is well within 
the limits of what may be considered a normal dispersion, which 
indicates that the conditions remained constant for this subject. 
Such individual variations as the coefficient for the equal judg- 
ments of the 88 gm. weight [1.568] or that for the equal judg- 
ments of the 104 weight [0.699] ma y be accounted for by the 
chance variations of the results. The values of the coefficient of 
divergence are dependent, to a certain extent, upon the size of 
the sum of the deviations. It will be noted that all of the averages 
for subject I are outside of the limits of what must be considered 
an over normal dispersion. 

For subject II, on the other hand, the total average is 1.412, 
which indicates a somewhat overnormal dispersion. All of the 
averages but one, the average for the 92 weight, are higher than 
the corresponding values for subject I. Four of the averages for 
subject II are above the limit for the over normal dispersion and 
two others are just below it, so that they almost certainly indicate 
the same tendency. This result indicates that a change in the con- 
ditions which determine the probabilities of the different judg- 
ments has taken place and we will have to consider what this 
change may have been. 

The passing of a certain judgment may be dependent upon 
either the physical conditions of the experiment or upon the 
psychophysical make up of the subject, or to a complex of both 
influences. The physical conditions of the experiment remained 
as constant as it was possible to control them, hence the variations 
must have had their origin within the subject himself. We may 
liken the psychophysical make up of an individual to our example 
of the black and white balls in an urn, in which those psycho- 
physical influences that control the passing of a certain judgment 
are likened to the balls. Now if these influences remain constant 
we will obtain a normal dispersion as in the first example where 



METHOD OF CONSTANT STIMULI 25 

the relative frequencies of the white and black balls in the urns 
were the same. The normal dispersion for subject I would indi- 
cate such a constancy of conditions. The overnormal dispersion 
in the results of subject II indicates an influence which was at 
work in this subject but not in subject I, who showed approxi- 
mately a normal dispersion. The changes in subject II may be 
likened to the case where the relative frequencies of the black and 
white balls varied for the different urns, and it seems to be an 
obvious idea to see this influence in the progressive practice ac- 
quired during the experiments. It will be remembered that 
subject I was very highly trained in the lifting of weights; while 
subject II had only one day's training at the start of the experi- 
ment; just enough to enable his hand movements to become 
somewhat automatic. It is, therefore, very likely that the changes 
in the conditions, as indicated by the coefficients of divergence, 
were due to the effect of practice. Similar results were found by 
Urban [Psych. Massmeth., Arch. f. ges. Psych., XV, p. 283], 
where it was found that the subjects with the most considerable 
training showed very nearly a normal dispersion. 

The changed conditions, as indicated by the coefficient of diver- 
gence greater than unity, may be due to a complex of four 
influences : first, a change in the sensitivity ; second, to an uncon- 
scious learning of the order of the stimuli ; third, to fatigue ; and 
fourth, to the effect of practice. Practice itself is a complex of 
two elements that are, however, closely related and dependent 
upon one another; first, the acquiring of the automatic movements 
of the hand and by this the direct effect of the elimination of the 
space error and constancy of the time error. Second, the direct- 
ing of the entire attention on the judgments, when it is no longer 
necessary to direct some of it upon the hand movements. 

It is possible that there may be a certain training of the sensi- 
tivity in the same way that we may have training in a muscle, 
bringing with it increased efficiency. The sense organs employed 
in this experiment, the end organs of touch in the hand and the 
free nerve endings in the wrist and the forearm, are end organs 
constantly in use. So it is reasonable to believe that they are nor- 
mally at a high state of efficiency so that this factor of the training 
of these sense organs cannot be of considerable extent. 



26 SAMUEL W. FERNBERGER 

The influence of fatigue could not have been great, as during 
the entire time of the lifting, we endeavored to eliminate this 
factor by resting the subject after every 5-7 revolutions of the 
table. Besides, the passing of a judgment on the intensities of 
two weights where the differences were as small as in this experi- 
ment, would be impossible in a state of fatigue, since in that case, 
the judgment would really become nothing but a guess. But even 
if we have present some influence of fatigue, this would be practi- 
cally eliminated, since we are comparing groups of 100 experi- 
ments, each of which represent three or four days' experimenta- 
tion. Thus there would be several places in the series, where the 
subject was fresh and as many places where he was fatigued, and 
these influences would tend to cancel one another. 

It seems a reasonable expectation that the effect of practice 
would be greatest in the beginning of the experiments and that 
the conditions should remain constant after a certain perfection 
is attained. In order to test this supposition, the coefficient of di- 
vergence was calculated for subject II for the last 13 groups, 
omitting the first group of 100 experiments. The amount of work 
in this second set of calculations, can be greatly reduced by deriv- 
ing the new sum of the squares of the deviations from the values 
already ascertained. This step, which is the one that requires 
the most time in the calculations, may be accomplished by applying 
the formula 

2 v\ = 2 v 2 + nd — (A 1 — a) 1 . 
In this formula, n represents the number of groups ; d the differ- 
ence between the average of the 14 groups and the average of the 
1 3 groups, which latter is represented by A 1 ; and finally, a stands 
for the omitted result. 

Table VI, which has the same form as table V, gives the new 
values of the coefficient of divergence for subject II for the last 
13 groups only. It will be noticed that, although the individual 
values change, the total average for the two sets of calculations 
for this subject are almost identical. This would indicate that 
the changed condition, whatever it may be, did not have its great- 
est effect in the first series. Where an individual coefficient in 
table VI is smaller than the corresponding value in table V, it 



METHOD OF CONSTANT STIMULI 



27 





Subject II 


Stimulus 


Lighter 
Judgments 


Equal 
Judgments 


Heavier 
Judgments 


Average 


84 
88 
92 
96 
104 
108 


1.180 
2.473 
1. 107 
1.802 
1.492 
I.I9S 


1.223 
I.55I 
0.953 
1.440 
2.051 
1.047 


1.306 
0.906 
1.265 
1.236 
2.282 
1.262 


1.236 

1643 
1. 108 

1-493 
1.942 
1.168 


Average 


1.542 


1-377 


1.376 


1-432 



Table VI 

indicates that the deviation in the first series was large. The 
opposite holds for the cases where the terms in table VI are larger 
than those in the former calculation. It will be seen, however, by 
the averages that, on the whole, these variations almost cancel 
one another. If the over normal dispersion was due to the effect 
of practice, we may know that this was not as rapid in the first 
series as might have been expected, but that it was gradual 
throughout the experiment. 

A further treatment of the coefficient of divergence strengthens 
the belief in the importance of the concentrating of the attention 
upon the judgments. If the averages for each intensity of the 
comparison stimulus for both subjects are averaged, we obtain the 
values 



84 


— 


1. 180 


88 


— 


i-4i5 


92 


■ — 


1-295 


96 


— 


1.402 


104 


— 


1.329 


108 


— 


1. 116 



It will be noticed that for the extreme values of the comparison 
stimulus, the averages of the coefficients of divergence are small 
and that, with the exception of the average for the 92 weight, 
they gradually rise toward the central values. Considering the fact 
that these are averages of the results of only two subjects, it is 
remarkable that this course should be broken at only one place. 
The explanation of this set of values seems to lie in the fact that it 



28 SAMUEL W. FERNBERGER 

takes less concentration of the attention to judge the difference 
between an 84 or a 108 gram weight and a standard weight of 100 
grams; than it would between the same standard weight and a 
comparison stimulus that was but little different from it. So at 
the start of the experiment, subject II could give enough atten- 
tion to the judgments on the extreme intensities of the comparison 
weight to have these judgments fairly accurate. It required, 
however, so much more attention to give correct judgments on the 
central values that, at the start of the experiment, these were 
judged much more inaccurately. Later when more attention could 
be given to the judgments, the central values could be judged more 
accurately, while there could be little change in the extreme 
values, as they had been judged with a fair degree of accuracy 
from the beginning. Thus there was greater variation for the 
central values than for the extreme ones, and this fact is shown 
by the size of the coefficients of divergence for the different in- 
tensities of the comparison stimulus. 

Although the two elements of the automatic movements of 
the hand and the increased attention on the judgments are inter- 
related, still it does not imply that when the automatism is per- 
fect, that we have at once a maximal attention on the judgments. 
It is our belief that for subject II, the hand movements became 
automatic very early in the experiment and this led us to calculate 
the coefficient of divergence for this subject for the last 13 groups 
of experiments. We found, however, but little change in the value 
obtained from that corresponding value for all of the experiments. 
This would indicate that the element of practice of the automatic 
hand movements is not of very great direct importance. This ele- 
ment, however, is fundamental for that of the proper direction 
of the attention upon the judgments, as the latter influence implies 
that the automatisms have reached a degree of perfection. 

The coefficient of divergence merely shows that a change in the 
conditions has taken place. It furthermore, indicates the nature 
of this change but, by no means, can it be taken as conclusive 
evidence. The examination of the constants of the psychometric 
functions of the two subjects will give us a chance to study this 
variation a little more closely. The psychometric functions for 



METHOD OF CONSTANT STIMULI 29 

the heavier judgments gives the probability of a heavier judg- 
ment as function of the intensity of the comparison stimulus. 
Similarly the psychometric functions for the lighter judgments 
gives the probability of a lighter judgment; and for the equal 
judgments gives the probability of an equal judgment, as func- 
tion of the intensity of the comparison stimulus. If the psycho- 
metric functions with all their constants are given, we are able to 
calculate the probabilities of the different judgments for every 
intensity of the comparison stimulus. We may represent the 
course of the psychometric functions graphically, by constructing 
the comparison weights on the abcissa and the corresponding 
probabilities of the different judgments as ordinates. The curve 
representing the psychometric functions for the lighter judgments 
will set in with high values [close to unity], and will drop at 
first slowly, then more rapidly and eventually it will approach the 
abcissa asymtotically. The psychometric functions for the heavier 
judgments will have just the opposite course, setting in with low 
values and attaining the values close to unity for the high intensi- 
ties of the comparison stimulus. The curve for the psychometric 
functions of the equality judgments starts with low values, then 
rises to a maximum and finally falls off very rapidly. A diagram 
of this kind enables us to see the variations of the probabilities 
of the different judgments at a glance. 

It would be the same if we were to get an analytic expression 
to express this set of facts. The choice of such a mathematical 
formula will be in the nature of a hypothesis about the psycho- 
metric functions. The one chief requisite of such an expression 
is that it fits the facts of the observed frequencies. Several such 
hypotheses can be advanced but they do not fit the facts of lifted 
weights, for example, as was shown by Urban in regard to the 
Lagrange formula [Method of Constant Stim., Psych. Review, 
XVII, p. 234]. The *(y) hypothesis recommends itself, how- 
ever, by its simplicity and the fact that it is known by large expe- 
rience. This experience has also shown that the *(y) hypothesis 
comes very near the truth in so far as lifted weights are concerned. 
This hypothesis underlies the method of constant stimuli, which 
is essentially nothing else but the determination of the quantities 



30 SAMUEL W. FERNBERGER 

h and c, two values upon which the form and position of the curves 
of the psychometric functions depend. The greater the value of 
h, the steeper the curve is and thus the h exerts an influence 
upon the form of the curve. The influence of c upon the position 
of the curve is such that a larger c means the shifting of the entire 
curve to the left. The essential feature of the *(y) hypothesis 
is that only the values of the extreme judgments are calculated. 
The values of the equality judgments are found by the difference 
from unity of the sum of the probabilities of the heavier and 
lighter judgments. The extreme judgments are those which are 
greater or less. The intermediate judgments admitted in any 
study must be odd in number and in this study it was limited to 
one, the equality judgment. The *(y) hypothesis consists in the 
supposition that the psychometric functions of the smaller judg- 
ments are represented by expressions of the form 

v =y 2 [i—* (h lX — Cl )], 

and for the greater judgments by 

v i = y 2 [i +*(h 2 x — c 2 )j. 

In these equations x represents the intensity of the comparison 
stimulus and * has the sign of its argument. If we substitute 
the relative frequencies of one of the extreme judgments, e. g. of 
the lighter judgments, for the values of the comparison stimulus 
used in our experiments, we obtain a series of equations of the 
form 

P84 = ^ [i—* (hx 84 — c)] 

Ps8 = y* [i — * (h x 88 — c)] 



from which we have to determine the unknown quantities h and 
c. It is not possible to solve these equations in this form and so 
they are changed to read 

2p 8 4 — i 
* (h x 84 — c) = 



2 

2p88 — I 



* (h X 88 — C) = 



METHOD OF CONSTANT STIMULI 31 

from which the arguments (hx — c) may be determined, which 
is the form to be used in our observation equations. The relative 
frequencies for all the judgments of all the six comparison stimuli 
are found in tables III and VI. The values of *(y) correspond- 
ing to these relative frequencies are found either in a table of the 
probability integral or in the so-called fundamental table for the 
method of right and wrong cases that was calculated by Fechner. 
It was found that many of the values in Fechner's original table 
were wrong in the last decimal place because the tables of the 
probability integral that were in Fechner's possession were not 
complete. With the appearance of complete tables of the proba- 
bility integral by Bruns [JVahrscheinlichkeitsrechmmg und 
Kollektivmasslehre., 1906] , the fundamental table was recalculated 
by Urban [Method of constant stimuli, Psych. Rev., XVII, p. 
251]. This situation was not understood by Brown [Mental 
Measurement, p. 134, 191 1], who prints this table and states that 
it was calculated by Bruns and quoted by Urban. 

These determinations of unknown probabilities, however, are 
not exactly correct and unless these differences are allowed for, 
certain discrepancies will appear in the results. To eliminate 
these errors, each observation equation is given a weight, which 
is in relation to the probable errors of the observed relative fre- 
quencies. G. E. Muller was the first to see that these observation 
equations are not of the same weight, but did not arrive at the 
correct formula. Urban took up this analysis and published a 
table of these weights, calculated by another formula [Psych. 
Review, XVII, p. 253, and Arch. f. ges. Psych., XVI, p. 371. 
also quoted by Brown, Mental Measurements, p. 135]. 

The values for 7 are substituted in the observation equations 
and it is found to be convenient to write the weight or P after 
each equation. Thus we obtain an observation equation and a 
weight for each of the six comparison stimuli used, in this form 

hx ± — = 7! with weight P x 

hx 2 — c = y 2 with weight P 2 



hx 6 — c = y 6 with weight P 6 . 
This gives an over determined set of equations for the deter- 



3,2 SAMUEL W. FERNBERGER 

mination of the constants h and c. The solution of this set, by 
the Method of Least Squares, gives the most probable values of 
the unknown quantities. For the purposes of calculation an 
adding machine was found to be invaluable. After some practice, 
it became possible, with the help of this machine, to effect the solu- 
tion of such a set of observation equations in one and a quarter 
hours, which is considerably less than the time required to do the 
same calculation even with the help of logarithms. 1 The calcula- 
tions are very easy but rather lengthy so that it is impossible to 
be sure of their correctness unless the whole work is arranged 
systematically [cf. Urban, Arch. f. ges. Psych., XVI, pp. 375- 
377, and Wirth, Psychophysik, pp. 210-214]. The scheme used 
for this purpose is a modification of the Gaussian method for 
solving a system of equations by the method of least squares. 
The first step for this solution consists in setting up the normal 
equations which have the form : 

[xxP]h — [xP]c = [xyP] 
— [xP]h+ [P]c = — [yP]. 

These normal equations require the calculation of the sums of 
the products xP, xxP, yP, and xyP. The sum of the P is found 
directly by addition. We obtain the products xP by multiplying 
each P by its corresponding x; their sum is designated by [xP]. 
The values xxP are obtained by multiplying each xP by the cor- 
responding x, a multiplication which is performed on the adding 
machine in two steps without clearing the machine. The calcula- 
tion of [yP] and [xyP] is similarly arranged. The products yP 
are formed first and then used for the calculation of xyP without 
clearing the machine. 

These six sums are all the values that are necessary for the 
setting up of the normal equations for h and c. In order to check 
the correctness of these results, three other values are calculated. 
A quantity s is defined as the algebraic sum of all the coefficients 
of the observation equations, e. g. by 

s = x — I — y 

1 Since the writing of this paper. Urban (Hilfstabelhn fur die Konstanz- 
methode. Arch, fur die ges. Psych. Bd., XXIV, 2-3 Heft, pp. 236-243) 
has printed a set of tables which are of great assistance in this kind of work. 
By means of these tables a set of equations can be solved in little more than 
half an hour. 



METHOD OF CONSTANT STIMULI 33 

notice being taken of the sign y. Multiplying this equation 
by P we obtain 

sP = xP — P — yP. 

If these products are formed for every observation equation and 
added we obtain 

[sP] = [xP] — [P] — [yP]. 

The terms on the right side of this equation are needed for setting 
up the normal equations and the calculation of the sum [sP] 
gives us a thoroughgoing check of our results for the other three 
sums. As all the decimal places were retained in the terms of 
this equation it must be expected to solve exactly. Multiplying 
the equation for s by xP gives 

xsP = xxP — xP — xyP 

and by adding up these relations for all of the values of x we 
obtain 

[xsP] = [xxP] — [xP] — [xyP]. 
The terms on the right side of the equation are the sums which 
are used in the normal equations and the sum [xsP] affords a 
check on our calculations of these sums. Our calculations were 
arranged in such a way as to require this check to tally to the third 
decimal place. The check worked out in this way is a great sav- 
ing of time as it requires only the calculation of three sums [s], 
[sP] and [xsP] ; the s being calculated almost directly. The 
only other check would be a recalculation of the other five quanti- 
ties. This would be not only more laborious but furthermore in 
such a recalculation, it would be quite possible to repeat an error 
that had been made before. 

Tables VII and VIII give these sums for the lighter and heavier 
judgments respectively for subject I and tables IX and X are 
similarly constructed for subject II. The first columns give 
the number of groups of the hundred experiments into which the 
results were divided. The succeeding columns give in order the 
values of [P], [xPj, [xxP], [yP], [xyP], [sP] and [xsP]. 
On account of the lack of space these tables are somewhat reduced, 
by omitting one decimal place for the values [xP], [xxP], [yP], 
[xyP] and [sP] ; and two for the quantity [xsP]. All the 



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38 SAMUEL W. FERNBERGER 

checks are fulfilled so that we are sure that no mistake has been 
made in the calculation of these sums. 

These values are substituted in the normal equations which 
must be solved for the two unknown quantities h and c. It is 
found to be simplest in most cases to solve for h directly, and then 
substitute the value found, in the normal equation for c; as in 
this equation the terms are smaller. A check on this part of the 
work is effected by substituting the values obtained for h and c 
in the normal equation for h and observing- whether the equation 
proves. This check will not come out exactly as only four 
decimal places are retained in the calculations, but we placed the 
arbitrary limit that the difference between the values of [xyP] 
obtained should not be greater than one per cent of that value. 

After we have obtained the values of h and c, the calculation 
of the threshold is very simple. The threshold in the direction 
of increase is defined as that point in the curve of the heavier 
judgments where the probability of a heavier judgment is J4. 
Similarly that place in the curve for the lighter judgments where 
the probability of a lighter judgment is J^, defines the threshold 
in the direction of decrease. At such a point, according to the 
*(y) hypothesis the value of y = o. Then 

o = hx — c. 

Then we obtain the formula 

c 
S = — 
h 

which is used for the calculation of the threshold in the direction 
of increase if the h and c for the heavier judgments are used. 
If the constants for the lighter judgments are substituted in the 
formula, the S will be the threshold in the direction of decrease. 
Tables XI and XII give the h, c and S values for subjects I and 
II respectively. Opposite the numbers for the series, given in 
the first columns, will be found the values of h, c and S for first 
the lighter and then the heavier judgments. The columns headed 
Si contain the threshold in the direction of decrease and those 
headed S 2 give the threshold in the direction of increase. 







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METHOD OF CONSTANT STIMULI 41 

By a close study of these two tables [XI and XII] we can dis- 
cover many facts that were only hinted at in the results of the 
coefficient of divergence. We will consider the effect of the un- 
conscious learning of the order of presentation of the stimuli. 
The order of the series was changed after the first, fifth and thir- 
teenth groups of 100. If there was any effect of this knowledge 
of the series, we should expect to find that the h for the group 
immediately before the change would be larger than the h in the 
group following. As has been noted above, the h controls the 
steepness of the curves and with a knowledge of the order we may 
expect a greater accuracy in the judgments. An examination of 
the tables will show that for subject I, the h's for the groups 
immediately after these changes are slightly smaller for the most 
part, than those of the series just preceding, but this is not true 
for subject II. This would indicate that for the latter, the influ- 
ence of the knowledge of the order was of no consequence. For 
subject I this knowledge may have had some effect upon the re- 
sults, but as the differences are small this effect was not 
considerable. 

For the convenience of studying the effect of practice, it was 
deemed advisable to plot the four curves of the course of the h's 
shown in figures I and II. Upon the abcissa are laid off the series 
in order; while the ordinates represent the values of h. These 
curves are strikingly different in form when we compare those 
of the different subjects; but for the the same subject they are 
quite similar in character. Although all the curves show unsys- 
tematic variations, the curves for subject I start with values that 
are fairly high and the general trend is upward. This is particu- 
larly noticeable in the curve h 2 for subject I. The curves hj and 
h 2 for the other subject start with comparatively small values and, 
for the first four groups with h 1; and the first six groups for 
h 2 , they show a very rapid rise. Both curves then fall from this 
maximum, and from the seventh series on, show a very gradual 
upward tendency. 

For reasons stated above we have ruled out the influence of the 
knowledge of the orders as being negligible. Changes in sensi- 
tivity, due to the weather conditions, the tone of the subject and 



1500 
75 
50 
25 

1400 
75 
50 
25 

1300 
75 
50 
25 

1200 

75 

50 

25 

1100 

75 

50 

25 

1000 



Curves for the 


values of hi. and hz. 




for 5ubj 


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I 



3 4 5 6 7 8 9 10 II 12 13 14- 




tO it 12 13 14 



44 SAMUEL W. FERNBERGER 

like influences may account for the unsystematic variations or 
the chance irregularities in the curves. But we have still to ac- 
count for the systematic tendencies that give the curves for the 
two subjects their distinctive character. We have eliminated all 
of the factors that could influence our judgments except that of 
practice; in fact the curves for subject II might be taken for typi- 
cal practice curves. 

This agrees with the facts regarding the subjects themselves, as 
it will be remembered that subject II was absolutely untrained at 
the start of the experiment; while subject I was highly trained, 
due to his having acted as subject in previous experiments. The 
values of h start high for the latter, but, in spite of all his former 
practice, the curve for subject I still shows a gradual rise. The 
curves for subject II indicate, by their rapid rise in the first six 
series, that during this time the subject became practised in the 
method of lifting the weights. This means that the hand move- 
ments became so automatic that the space error was entirely elimi- 
nated. This may perhaps need some explanation. At the start 
of the experiment there was a tendency on the part of subject II 
to turn his hand slightly in the direction from which the weights 
approached, and thus a space error, even though a slight one, was 
committed. This, however, was subsequently eliminated when 
the hand movements of the subject became automatic, and there- 
fore accurate in regard to movement in space. The hand move- 
ments improved in regularity as to time and so the time error be- 
came more constant as the experiment proceeded. There was 
still another effect of the perfecting of the automatism, since when 
perfected it no longer requires any of the attention of the subject, 
and this then, can be entirely focused on the judgments. In this 
connection, it will be remembered, that the results of the calcula- 
tion of the coefficient of divergence argue for the same notion. 
The drop in the curves after the maximum had been reached 
cannot be accounted for unless the maximum itself is caused by 
a chance variation that accentuates it. If a similar curve be con- 
structed for subject I from the results of the former experiment 
in which he acted as subject, it will show a similar rapid rise in 
the early series. Several years have elapsed between that experi- 



METHOD OF CONSTANT STIMULI 45 

merit and this one. If the sense organs had been trained due to 
the practice several years ago, to a higher state of efficiency; it 
is reasonable to believe that due to the disuse during the interval 
between the experiments, they would have degenerated somewhat 
to the condition in which they were before they had received any 
training at all. The rise in the curves for subject II are not very 
great and this would argue against any considerable education 
of the sense organs. 

It must further be remembered that the measure of the sensi- 
tivity of the sense organ is not to be found in the abruptness of 
the curve of the psychometric functions, h, but by the interval 
between the two thresholds. This is the basis of the practicability 
of the method of constant stimuli. An examination of Tables XI 
and XII will show that the thresholds remain fairly constant. 
These values show chance variations but it is impossible to dis- 
cover systematic tendencies in their course. They remain con- 
spicuously constant for subject I. Thus the training of the sense 
organ, which might directly effect the sensitivity seems to have 
little influence, if any, as the sensitivity remains very constant 
and at least shows no systematic variations. This is an important 
consideration, because if practice effected the measure of sensi- 
tivity, the method of constant stimuli would have to be abandoned 
as a practical means of arriving at that value. It will be noticed, 
however, that, although the values of the h and c vary considera- 

c 
bly, — the threshold, remains practically constant, 
h 

The characteristics of the curves for the two subjects agree 
with the values of the coefficient of divergence obtained for the 
same set of results. For subject I we obtain a coefficient of diver- 
gence that approximates unity more closely than that for subject 
II, and the curves of the former show not only smaller unsyste- 
matic variations, but smaller systematic ones as well. Our ex- 
periments were probably not extended enough to show long 
periodicity and the results were not regular enough to reveal short 
periodicity. 



46 SAMUEL W. FERNBERGER 

It then seems that the important variations in the curves of the 
h and c values are : 

1. Those of an unsystematic or chance character, that are due 
to the condition of the subject and similar chance influences. 

2. Variations of a systematic nature that seem to have their 
origin principally as the result of practice which allows a concen- 
tration of the attention primarily upon the judgments, when the 
hand movements become so automatic that they no longer require 
attention. 

3. This effect of practice is not evenly distributed throughout 
the experiment, but for an untrained subject is comparatively 
great in the first few hundred experiments. 

4. This effect of practice, however, does not seem to effect the 
real purpose of the experimentation, which is to ascertain the 
sensitivity of the subject, as the ratios that define the measure of 
sensitivity remain practically constant and show no systematic 
variations. 

We will turn now from the study of the effect of practice, as 
shown in this set of results, to a consideration of the results them- 
selves. The real purpose of the psychophysical methods is to 
ascertain the sensitivity of the subject. The problem is how much 
larger or how much smaller a comparison weight must be from 
the standard weight until a difference can be perceived between 
them. This statement implies that there is a distance on each 
side of the standard weight, within the limits of which a compari- 
son stimulus, although physically lighter or heavier than the 
standard weight, will not be judged lighter or heavier with 
a probability of 0.50. This distance is called the interval of uncer- 
tainty and is defined as the difference between the two thresholds. 
The measure of sensitivity is one half this interval. For example, 
if we have a standard weight of 100 grams and our interval of 
uncertainty is 4 grams, the measure of sensitivity will be 2 grams. 
This means that either a weight of 102 or of 98 grams will be 
distinguished from a weight of 100 grams with a probability 
of 0.50. 

But these are not all the considerations that are necessary to tell 



METHOD OF CONSTANT STIMULI 47 

the sensitivity of our subject. It will be remembered that our 
results are effected by constant errors. Two weights of ioo 
grams if lifted in either different spacial or temporal relations 
will not be subjectively equal. In our experiments the space 
error was eliminated by the turning top table, but the time error 
was present, although it was controlled by the regular beats of 
the metronome. We must now find the point of subjective equal- 
ity, which is that weight which, under the conditions of our experi- 
mental procedure, will be subjectively equal to our standard 
weight. This is found by applying the formula 

Ci +c 2 



K + h 



When this point of subjective equality has been ascertained, we 
can find the influence of the constant errors almost directly, by 
finding the difference between the point of subjective equality 
and the standard stimulus. We will consider how this effects 
our example given above. Suppose we find that the influence 
of the time error is — 3 grams. In this case our point of subjec- 
tive equality is 97 grams. The measure of sensitivity remains the 
same, however. Thus a restatement of our results indicates that 
without the influence. of the time error, weights of 95 or 99 grams 
can be distinguished from one of 97 grams. 

Table XIII is a double table containing the values for our 
subjects of these quantities we have just discussed. For each 
subject will be found in order the values of the interval of uncer- 
tainty ; the point of subjective equality ; and the time error. These 
will be found opposite the number of each of the groups of 100 
experiments into which our results were divided. At the bottom 
will be found the averages for each of the columns. 

It will be noticed that the interval of uncertainty remains very 
constant for both subjects although the values are by no means 
identical. The variations are obviously due to changes in the 
sensitivity of the subject. These are the changes that are caused 
by the variations in the psychophysical makeup of the individual. 
There does not seem to be any correlation between the character of 



48 



SAMUEL IV. FERNBERGER 



these changes for the two subjects. We see that the averages of 
the interval of uncertainty are almost identical for the two sub- 
jects. This is of course a matter of chance and merely indicates 
that the sensitivity of the two subjects happened to be almost 
the same. 

The time error was a negative one in the Fechnerian sense ; that 
is, the second weight lifted was relatively heavier. This means 
that the point of subjective equality is smaller than the standard 
weight. Thus, on the average, for subject I for our arrange- 
ment of the experiment, a standard weight of ioo grams would 
be subjectively equal to a comparison weight of 97.31 grams. 
It will be noticed that the point of subjective equality is practically 
midway between the two thresholds. This indicates that the 
amount that must be added in order to perceive a difference be- 
tween the two weights is approximately the same as the amount 
that must be subtracted in order to just perceive a difference. For 
our calculations the point of subjective equality remained com- 
paratively constant for each subject, but there is considerable 
difference [one gram on the average] for the two subjects. The 
time error for subject I seems to become less as the experiment 





Subject I 


Subject II 






Point of 






Point of 




Number 


Interval of 


Subjective 


Time Error 


Interval of 


Subjective 


Time Error 


of Groups 


Uncertainty 


Equality 




Uncertainty 


Equality 




I 


5-49 


96.83 


— 3-17 


4.90 


95-30 


— 4-70 


II 


5.92 


96.97 


— 3.03 


4.80 


97.09 


— 2.91 


III 


4.98 


96.76 


— 3-24 


3-23 


96.80 


— 320 


IV 


4.85 


97.64 


— 2.36 


370 


96.17 


— 3.83 


V 


5.52 


96.76 


— 3.24 


3.18 


96.24 


— 3-76 


VI 


5-42 


96.87 


— 3-13 


4.28 


95-72 


— 4-28 


VII 


4-83 


97-34 


— 2.66 


4.98 


96.29 


— 371 


VIII 


4.42 


97.12 


— 2.88 


5-70 


95.42 


-4.58 


IX 


4.26 


97.21 


— 2.79 


5-47 


95-50 


— 4-50 


X 


3.84 


98.06 


— 1-94 


4.80 


96.16 


— 3-84 


XI 


5-52 


9747 


— 2.53 


5-72 


95-67 


— 4-33 


XII 


5.08 


9749 


— 2.51 


5-84 


97.62 


— 2.38 


XIII 


4.80 


98.32 


— 1.68 


6.13 


96.95 


— 3-05 


XIV 


548 


97.5o 


— 2.50 


5.25 


97-43 


— 2.57 


Average 


503 


97.31 


— 2.69 


4.86 


96.31 


— 3-69 



Table XIII 



METHOD OF JUST PERCEPTIBLE DIFFERENCES 49 

proceeds. The physical conditions remained the same throughout 
the experiment as this constancy is attested by the fact that there 
are no systematic variations in the course of the time error for 
subject II. This systematic change, found in the results of sub- 
ject I is not very great and thus may be due to the chance arrange- 
ment of the results. If it is significant, however, it must be due 
to some change in the psychophysical organism. It is not due to 
an effect of practice, since on the basis of the discussion of prac- 
tice given above, we should expect to find an even stronger 
tendency in subject II, where it is entirely absent. 



IV. METHOD OF JUST PERCEPTIBLE DIFFERENCES 

The classical form of procedure of the method of just per- 
ceptible differences is to start with two stimuli at subjective 
equality and increase the comparison stimulus C by equal steps 
only up to the point where a difference is first noticed ; this point 
we call the just perceptible positive difference. We then start 
again, to obtain the just imperceptible positive difference, with the 
comparison stimulus C subjectively greater than the standard 
stimulus S ; then C is decreased until no difference is perceived 
between C and S. Again we start with C and S subjectively 
equal and decrease C until it is first judged less and thus obtain the 
just perceptible negative difference. Finally, to obtain the just 
imperceptible negative difference, we start with C subjectively less 
than S and increase C to the point where it is first impossible to 
perceive a difference. Then the just perceptible and imperceptible 
positive differences are averaged and we obtain the threshold in 
the direction of increase. The threshold in the direction of de- 
crease is found by averaging the just perceptible and imperceptible 
negative differences. This is the form in which this method was 
described by Fechner [Elemente der Psychophysik, I, p. J2\, 
Wundt [Phys. Psych., I, 1902, p. 475], Muller [Gesichtspunkt 
und die Tatschen der psychophysischen Methodik, 1904, p. 179], 
and Titchener [Exp. Psych. II, I, p. 56]. 



50 SAMUEL W. FERNBERGER 

Another form of the method of just perceptible differences is 
the one used in this experiment, which decreases the amount of 
experimentation considerably and has also other advantages. We 
start with C so much lighter than S that there is a very small 
probability of any but a lighter judgment, and then increase the 
comparison stimulus C by successive steps until there is a very 
small probability of any but a heavier judgment. The four 
differences, from which the thresholds are obtained, are thea 
picked out of the results and are defined in the following manner. 
The just perceptible negative difference is the greatest stimulus 
upon which a lesser judgment is passed. The just imperceptible 
negative difference is the smallest stimulus upon which a not- 
lighter — either equal or heavier — judgment is given. The small- 
est stimulus upon which a heavier judgment is given becomes 
the just perceptible positive difference. Finally, the greatest 
stimulus upon which a not-greater — equal or less — judgment is 
passed is the just imperceptible positive difference. Thus in 
one experimental operation we obtain all the values that in 
the other form of the method required four distinct series. 
After these differences are obtained, the thresholds are found 
in the same manner as in the other method. Besides the matter 
of expediency, there is another great advantage of this form of 
the method over the classical form. The original method requires 
that we start at the point of subjective equality; but this term is 
not defined and the method gives no indication as to what intensity 
of the comparison stimulus is to be used for it. In actual practice 
this method is handled in such a way that the comparison stimulus 
is used, as a starting point, on which an equality judgment is 
given. This definition, however, is very loose since a great 
number of comparison stimuli fulfill this requirement. The 
variation of the method now under consideration, avoids this dis- 
advantage by not making use of the notion of subjective equality 
at all. The fact that this variation gives satisfactory results indi- 
cates clearly that only the so-called upper and lower limits of the 
interval of uncertainty are of consequence for the determination 
of the sensitivity of the subject. From this it seems to follow that 



METHOD OF JUST PERCEPTIBLE DIFFERENCES 51 

the discussion as to whether the arithmetic mean or the geometric 
mean ought to be taken as representative of the point of subjec- 
tive equality, cannot be decided on the ground of this method; 
since it does not give any definition of that value whatsoever. 
[Wundt, Phys. Psych., I, 5th Ed., p. 477.] 

In order to avoid the so-called errors of expectation, a descend- 
ing series of just the reverse experimental procedure was taken 
for every ascending series, of the form that has been described. 
This influence of expectation may be twofold; either, if the sub- 
ject knows that he is in an ascending series, he may give a heavier 
judgment too soon, or that knowledge may bias him against 
giving such a judgment. Similarly he may give a lighter judg- 
ment too soon or not soon enough, if the subject knows that he is 
in a descending series. There is, however, a great difference of 
opinion as to the extent of the influence of expectation. For 
example, Wundt [Phys. Psych., I, 1902, p. 479, 491] believes that 
this tendency may be overcome by practice; and Fullerton and 
Cattell [Small differences] think that this influence is so great 
that they strongly recommend that the subject should have no 
knowledge of the type of series that he is judging. In many cases 
it is not possible to keep this knowledge from the subject. For 
example, in the present study, the subject could tell from the 
sound made by the bearings being placed in the cylinder, whether 
he was judging an ascending or descending series. In the old 
form of the experimental procedure of this method, it was not 
possible to keep this knowledge from the subject because, after the 
first judgment, he could realize which type of series he was judg- 
ing. On the other hand, Titchener [Exp. Psych. II, II, p. 128] 
states that with good subjects this expectation has no influence 
whatsoever. Whether this is true or not, the method of double 
procedure, used in this study, allows for this error if present. 
Since it is not possible to avoid this error, one tried to eliminate it, 
or at least, to minimize its influence by using ascending and 
descending series alternately. The errors being in opposite direc- 
tion in the series may be expected to cancel one another. 

We will now analyze the nature of the four differences upon 



52 SAMUEL W. FERNBERGER 

which this method is based. It was believed for a long time, that 
the method of just perceptible differences uses notions which are 
foreign to the error methods. Urban, however, has shown that 
the common source of both methods is to be found in the notion of 
the probability of a certain judgment. The just perceptible nega- 
tive difference may be defined as the largest stimulus on which a 
lighter judgment was given. This implies that none of the greater 
comparison stimuli were judged lighter, but that on this one a 
lighter judgment was given. Thus by definition of the other dif- 
ferences, the just imperceptible negative difference implies that 
all the smaller stimuli were judged lighter but this one was judged 
not-lighter. The just perceptible positive difference implies that 
all smaller stimuli were judged not-heavier while on this one a 
heavier judgment was passed. Finally the just imperceptible 
positive difference implies that, on all greater stimuli, a heavier 
judgment was passed but on this one a not-heavier. Now let q 
be the probability of a lighter judgment on a given intensity of the 
comparison stimulus. Then by definition, the probability of a not- 
lighter judgment will be 1 — q, as these probabilities are mutually 
exclusive. In this same way 1 i — p will be the probability of a 
not-heavier judgment, if the probability of a heavier judgment is 
p Now suppose we have a series of comparison stimuli r 1? 
r 2 , . . . r n arranged in the order of their intensity, in which r x 
is the heaviest, and we use this series for the determination of 
the just perceptible negative difference. Let us designate the 
probabilities of a lighter judgment on the first, second, . . . n com- 
parison stimuli by q 1} q 2 , . . . q n . There exists for each stimulus 
a certain probability that it will be obtained as a determination of 
the just perceptible negative difference, which we call O l9 Q 2 , 

• ■ • a . 

Q 1 = & ; 

£»-! = (i-?x)(i-?,) U-O*. . ; 

Q n = (l-ft)(l-fc) (1-^-xK • 

The probability that a stimulus will be obtained as a determina- 



METHOD OF JUST PERCEPTIBLE DIFFERENCES 53 

tion of the just imperceptible negative difference can be similarly 
analyzed. For the comparison stimuli r ls r 2 . . . r n we have 

Q\ = ?n ■ ?n-i 9 2 (1— ?0 ; 

Ql = 9 n ■ 9 n -i • • 9% (i— 9,) ; 

Ql-i = 9 n (i—9 n - l ) ; 
Ql = (l-O . 

The probabilities that the just perceptible positive difference will 
fall on the stimuli r l5 r 2 , • • • r n are expressed 

A = (l-AXl-A-0 (1— A)A ; 

P % = kl—pn) a—pn-i) (I" A) A 5 

p n -, = (i— a)/«-i ; 

P n =Pn ■ 

Finally the probabilities with which the different stimuli will 
be obtained as a determination of the just imperceptible positive 
difference for the stimuli r l5 r 2 , . . . r n are 

p\ = (i-A) ; 

^i = Ad-A) ; 

p \-x=Px -A — • • /«_ (l— A-0 ; 

Pl=Pi -A A-x(l— A) • 

We now turn to a discussion of the results of our experiments. 
It will be remembered that one of the comparison stimuli was 
changed systematically so that the results on it could serve as a 
determination of the thresholds of the method of just perceptible 
differences, and at the same time, the judgments on the other 
comparison stimuli enabled us to apply the method of constant 
stimuli; the 'relation of these two methods being the principal 
problem of this paper. Obviously there is no difference in the 
form of the individual judgments of these two methods, since they 
consist in the passing of a judgment of whether the comparison 
stimulus is subjectively lighter, equal or heavier than the standard 
stimulus. Our results of the method of just perceptible differ- 
ences are of two forms; observed values that were taken simul- 
taneously with those by the method of constant stimuli, and 
calculated values. One way of obtaining these calculated values 



54 



SAMUEL IV. FERNBERGER 





I 


II 


III 


IV 


V 


J VI 


VII 


VIII 


1 
1 IX X 


Total 

i 


84.80 




10 


10 


10 


10 


10 


10 


10 


10 


10 


100 


85.22 






















10 


85.64 




10 


















10 


86.06 






10 


10 














20 


86.48 




10 






10 












10 


86.00 












10 










30 


87.32 




10 


10 








10 








20 


87.74 


10 






10 








10 






30 


88.16 




10 














10 




20 


88.58 












10 | 








10 


30 


89.00 




10 




10 




10 










40 


89.84 




10 


10 
















10 


90.26 


10 












10 








20 


90.68 




10 






10 












20 


91.10 






10 


10 








10 






30 


91-52 




10 








10 






10 




50 


92.36 


10 


10 


10 


10 


10 










10 


40 


93-20 




10 










10 








20 


93-62 






10 
















20 


94.04 


10 


10 




10 




10 




10 






50 


94.88 




10 


10 




10 












20 


95.30 


10 




















10 


95-72 




10 














10 




30 


96.14 


10 




10 


10 






10 








40 


96.56 


10 


10 


















10 


96.98 


10 








10 


10 








10 


60 


97.40 


10 


10 


















10 


97.82 


10 






10 








10 






40 


98.24 


10 


10 


















20 


98.66 


10 




10 
















10 


99.08 




10 






10 




10 








30 


99.50 


10 






10 




10 






10 




50 


99-92 




10 


10 
















20 


100.76 


10 


10 


















10 


101.18 








10 


10 






10 




10 


50 


101.60 




10 








10 


10 








50 


102.44 


10 


10 


10 


10 














20 


103.28 




10 






10 








10 




30 


103.70 






10 
















10 


104.12 




10 


















10 


104.54 


10 






10 




10 




10 






50 


104.96 




10 


10 








10 








10 


105.38 










10 










10 


30 


105.80 




10 


















10 


106.22 






10 


10 














20 


106.64 




10 


















10 


107.06 


10 










I 10 






10 




40 


107.48 




10 


10 




10 




10 








10 


107.90 








10 








10 






40 


108.32 




10 


















10 


108.74 






10 


















110.00 


10 










10 








10 


30 


1 11.26 














10 




10 




20 


112.52 












10 










10 



METHOD OF JUST PERCEPTIBLE DIFFERENCES 55 

is that employed by Urban [Stat. Meth.], where the experimental 
data were in such a form that it was possible to apply both forms 
of mathematical treatment to the same set of results. This is 
not possible in the present study as a glance at the table below will 
convince one that not enough judgments were taken on each 
intensity of the comparison stimulus to enable us to apply the 
treatment of the method of constant stimuli. In this table are 
found the number of judgments passed on the intensities of the 
comparison stimulus for each series, and the last column gives 
the totals for each intensity used. 

The other way of approaching this problem is to take the 
results of the method of constant stimuli and calculate from them 
the results we are likely to obtain from the method of just per- 
ceptible differences. This is possible because the determination 
of the quantities h and c determines the entire course of the 
psychometric functions. We, therefore, are able to obtain the 
probabilities of all three judgments on any comparison stimulus, 
if the constants h and c for the lighter and heavier judgments are 
given. These probabiltiies are indeed, all that we need for the 
calculation of the method of just perceptible differences, and we 
therefore can find the most probable value of any one of the four 
differences. The conditions under which our results for the 
methods of just perceptible differences and constant stimuli were 
taken, were very much the same, if not entirely alike, and we 
therefore, may expect to obtain calculated results which agree 
with the observed results within the limits of accuracy obtainable 
in this kind of experiments. The calculation of the probabilities 
with which the different intensities of the comparison stimulus 
will be obtained as determinations of the different perceptible and 
imperceptible positive and negative differences is easy but rather 
long and somewhat complicated. So it is necessary to plan the 
whole work systematically in a manner which will now be 
described. 

The first step consisted in the collection of the material for 
the determination of the constants of the psychometric functions. 
The records, as they were taken during experimentation, were 
entered in tables, similar in form to those described above as the 



56 SAMUEL W. FERNBERGER 

records of first entry, but the records of each series of the method 
of just perceptible differences were kept separate. The relative 
frequencies of the judgments in this division of the results by 
series, are found in tables XIV and XV for subjects I and II re- 
spectively. These are similar in form to tables III and IV, with 
the exception that the numbers in the first column, instead of rep- 
resenting groups of ioo judgments, as in the former calculation, 
now indicate the series of the method of just perceptible differ- 
ences with which they were taken simultaneously. As has been 
pointed out above, the length of the different series of the method 
of just perceptible differences varied ; there being only seven judg- 
ments for series X ; while series II required the passing of twenty- 
nine judgments. Thus it is obvious that the number of judgments 
from which these relative frequencies were calculated vary in a 
similar ratio. 

From these relative frequencies, the constants for the method of 
constant stimuli, h and c, were calculated by the same method 
that has been described above. In order to show the steps of this 
calculation, tables XVI and XVII give, for the lighter and heavier 
judgments of subject I, the values of [P], [xP], [xxP], [yP], 
[xyP], [sP] and [xsP]. The corresponding values are found 
for subject II in tables XVIII and XIX. These four tables are 
similar in form to tables VII-X, which have been described above. 
The numbers in the first five columns give the coefficients for 
setting up the normal equations and the data of the columns [sP] 
and [xsP] show that all the necessary checks are fulfilled. Tables 
XX and XXI, which are similar in form to tables XI and XII, 
give the constants h and c for the lighter and heavier judgments 
for both subjects. The thresholds S x and S 2 are obtained in the 
same way as in the former calculation and are included in these 
tables. The series of the method of just perceptible differences 
were performed in haphazard order, so that the results that go 
to make up the values included in these tables, are from various 
parts of the experiment at different stages of practice, and we, 
therefore, must not look for the effect of practice in these constants. 
In fact the results that were taken at the end of the experiment 
in each series, where the practice was high, tend to cancel any 



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1 — 11 — I ^*° i^* ^ I — 11 — irS?S 

MM K^^i—ll— 1 



6o 

Subject I 



SAMUEL IV. FERNBERGER 





Lighter Judgments 


Heavier Judgments 


Series 


h, 


c, 


s, 


h 2 


c 2 


s 2 


I 


0.11214 


10.655 


95.02 


0.12064 


12.076 


100.10 


II 


0.1 1266 


10.705 


95-Q2 


0.12071 


12.050 


99.82 


III 


0.1 1354 


10.732 


94.52 


0.12315 


12.287 


99-77 


IV 


0.1 1883 


11.266 


94.81 


0.1 1088 


11. 121 


100.30 


V 


0.12186 


11.480 


04.20 


0.12373 


12.286 


09.30 


VI 


0.1 1265 


10.644 


9448 


0.12340 


12.332 


99-93 


VII 


0.1 1650 


10.958 


94.06 


0.12359 


12.246 


99.09 


VIII 


0.11833 


11.211 


94-74 


0.12528 


12.459 


9945 


IX 


0.132 13 


12.519 


9475 


0.12345 


12.323 


99.82 


X 


0.12429 


11.670 


93-89 


0.14430 


14.276 


98.93 



Table XX 



Subject II 





Lighter Judgments 


Heavier Judgments 


Series 


h. 


Ci 


s, 


h 2 


c 2 


s 2 


I 


0.1 1 824 


11.027 


93-26 


0.1 1 783 


11.659 


98.94 


II 


0.11210 


10.587 


04.44 


0.1 1 134 


11.018 


98.96 


III 


0.1 1295 


10.636 


94.16 


0.1 1297 


11.092 


98.19 


IV 


0.1 1 189 


10.491 


9376 


0.10936 


10.752 


98.32 


V 


0.1 1007 


10.263 


9324 


0.1 1875 


11.740 


98.86 


VI 


0.10147 


9.472 


93-34 


0.10204 


10.038 


98.37 


VII 


0.13067 


12.273 


93-93 


O.I 1000 


11.777 


98.96 


VIII 


0.1 1490 


10.831 


94-27 


0.1 1320 


11.208 


99.00 


IX 


0.12538 


11.810 


94.20 


0.13827 


13-597 


98.34 


X 


0.12024 


"■371 


9457 


0.10909 


10.805 


99-04 



Table XXI 

variation of the results at the beginning, where practice was low. 
Thus these values of the threshold show remarkably little varia- 
tion. The numbers under the headings S x and S 2 give the 
thresholds in the direction of increase and decrease by the method 
of constant stimuli, for the group of experiments that were made 
simultaneously with the corresponding series of the method of 
just perceptible differences. 

After having obtained the constants h and c, we can ascertain 
the probability of a heavier or lighter judgment for any intensity 
of the comparison stimulus, by the formula 
p = y 2 ± y 2 *(y) . 
This calculation is made in two steps ; as first y must be calculated 
by the formula 

7 = hx — c 
in which x is the value of the intensity of the stimulus for which 
we are calculating the probability. The value of *(y) is then 
looked in a table of the probability integral [Czuber. Wahr- 



METHOD OF JUST PERCEPTIBLE DIFFERENCES 61 

scheinlichkcitsrechniing, 1908, pp. 388-391]. This value of *(y), 
which takes the sign of its argument, is substituted in the formula 
given above and we obtain the probability of that judgment for 
the weight x. These values must be found for every intensity of 
the comparison stimulus that was used in the corresponding series 
for the method of just perceptible differences. 

In actual practice this calculation is very much simplified. If 
we desire the probabilities of the lighter judgments q, for series 
II, we first find y 1 for x = 84.80, this being our lightest compari- 
son weight. We then find the value of y for x equal to the 
amount of difference of the steps of this series. This value is 
subtracted from y x and we obtain y 2 or that of the next heavier 
comparison stimulus. Then y is subtracted from y 2 and we 
obtain y 3 for the next heavier comparison weight, and so forth. 
As our y values are only carried out to four places, corrections 
have to be made when necessary. A check on this step of the cal- 
culations is effected by finding the y of the heaviest comparison 
weight by means of the formula, and ascertaining if it coincides 
with the value obtained by the series of subtractions. These y's 
are then successively introduced into the formula for finding 
the probabilities, given above, and the different probabilities of 
the lighter judgment — q ■ — are obtained for all of the comparison 
stimuli used. The probabilities that a not-lighter — equal or 
heavier — judgment will be passed are obtained by subtracting 
the probabilities of a lighter judgment, in each case, from unity. 

From this series of the q's and (I — q)'s we may obtain the 
just perceptible negative difference and the just imperceptible 
negative difference, by means of the formulae given above. This 
calculation is much simplified by the use of logarithms. These 
are looked up for every value of q and 1 — q. Then the probabil- 
ity that our lightest weight will be obtained as a determination 
of the just imperceptible negative difference is Q 1 = (1 — q 2 ) 
and is found directly. Adding log q ± to log (1 — q 2 ) gives 
log Q 2 which is the probability that our next heavier comparison 
stimulus will be obtained as a determination of this difference. 
Then log q x is added to log q 2 and to their sum is added log 
( 1 — q 3 ) which gives Q 3 . To the sum of log q x and q 2 is added 
log q 3 and to this sum is added log ( 1 — q 4 ), which gives the 



62 



SAMUEL W. FERNBERGER 



probability that this difference will fall on the fourth comparison 
weight. This scheme of calculation is very simple and is learned 
rapidly ; its mechanism will be easily understood from the follow- 
ing example, where the formulae are found alongside part of one 
of our calculations. 



log Q x 



log q x 

log (i-qj 
log (q 2 ) 



log Q 2 = log (i-qj q 2 

log (i-qi) 
log (i-q 2 ) 



log (i-qi)(i-q 2 ) 
log q 3 

log Q 3 = log (i-qi)(i-q 2 ) q 3 

log (i-qi)(i-q 2 ) 
log (i-q 3 ) 



log (i-qi)(i-q 2 )(i-q 3 ) 

log q 4 



= -94448 — 3 

= .99616 — 1 
= .44716 1 — 2 


= -44332 — 2 
= .99616 — 1 
= -98767 — 1 


= .98383 - 1 
= .81558 - 2 


= -79941 — 2 
= .98383 - 1 
= 97063 — 1 


= .95446 — 1 
= .07700 — 1 



log Q 4 = log (i-qi)(i-q 2 )(i-q 3 ) q 4 — -03146 — 1 



In this way the calculation is reduced to a number of successive 
additions by carrying the sums of the terms that accumulate. 
These additions give us the values of Q and Q lt or the probabil- 
ities with which the just perceptible and imperceptible negative 
differences will fall on the different values of the comparison 
stimulus. A similar calculation with the h and c for the heavier 



METHOD OF JUST PERCEPTIBLE DIFFERENCES 63 

judgments gives the values of P and P 1} or the probabilities that 
the different intensities of the comparison stimulus will be found 
as a determination of the just perceptible and imperceptible posi- 
tive differences. Both sets of calculations for the heavier and 
lighter judgments, had to be performed for each of the ten series 
for both subjects. 

A check on this part of the calculations is effected by the fact 
that, as these are probabilities of mutually exclusive events, all 
the values in one set must add up to unity. This had to prove 
within a limit of 0.0001, which was the exactitude of our calcula- 
tions. It was found for the longer series that this calculation did 
not have to be carried through for all of the values of x, as fre- 
quently the probabilities of the differences became so small as to 
no longer affect the last decimal place. In the short series, 
on the other hand, it frequently happened that there was a 
remainder, and this had to be calculated as it was necessary 
for a check of the correctness of our work. This remainder 
has some significance since it gives the probability that the 
series of comparison stimuli r t , r 2 , . . . r n will be gone through 
without giving a determination of the quantity sought fbjr. 
Thus the remainder for the just perceptible positive difference 
gives the probability that no stimulus of the series will be obtained 
as a determination of this quantity or — what is the same — that 
no heavier judgment will be given on any of the stimuli used. 
This remainder has the form of a product of probabilities and 
obviously becomes very small if any factors are small. Thus in 
our short series there was greater likelihood of having larger 
individual probabilities and we find a tendency for greater 
remainders to occur in the short series rather than in the long 
ones. This is the reason of the rule that the series of comparison 
stimuli should be extended so far that we will have a very high 
probability of obtaining only heavier judgments on the largest 
comparison stimulus, and only lighter judgments on the smallest 
stimulus used. It is, therefore, not desirable to work with series 
in which the remainders are of at all considerable size. None of 
the remainders of our series are large enough to invalidate our 
results in any way. 
Tables XXII-XXXI give the results of these calculations for 



Series I 



Comparison 














Stimulus 


q 


Q 


Q 1 


P 


p 


P 1 


84.80 


0.947 


O.OOOI 


0.0526 


0.005 


0.0046 




8774 


0.876 


0.0003 


0.1 177 


0.018 


0.0174 




90.26 


0.774 


0.0014 


0.1871 


0.047 


0.0456 




92.36 


0.663 


0.0034 


0.2164 


0.093 


0.0869 




94.04 


0.561 


0.0066 


0.1870 


0.151 


0.1273 




95-30 


0.482 


O.OIIO 


0.1239 


0.206 


0.1482 


O.OOOI 


96.14 


0.429 


. 0.0172 


0.0658 


0.250 


0.1422 


0.0001 


96.56 


0.403 


0.0270 


0.0295 


0.273 


0.1168 


0.0004 


96.98 


0.378 


0.0406 


0.0124 


0.297 


0.0925 


0.0013 


97.40 


0.353 


0.0586 


0.0049 


0.322 


0.0705 


0.0038 


97.82 


0.328 


0.0809 


0.0018 


o.349 


0.0516 


0.0106 


98.24 


0.304 


0.1082 


0.0006 


0.376 


0.0362 


0.0269 


98.66 


0.282 


0.1392 


0.0002 


0.403 


0.0242 


0.0640 


9950 


0.234 


0.1548 


O.OOOI 


0.459 


0.0165 


0.1261 


100.76 


0.181 


0.1437 




o.545 


0.0106 


0. 1948 


102.44 


0.1 19 


0.1075 




0.655 


0.0058 


0.2255 


104.54 


0.065 


0.0630 




0.776 


0.0024 


0.1889 


107.06 


0.028 


0.0278 




0.883 


0.0006 


O.I 120 


110.00 


0.009 


0.0088 




0-954 


O.OOOI 


0.0456 


2 




I. OOOI 


1. 0000 




I.OOOO 


I. OOOI 


R 




0.0000 


0.0000 




0.0000 


0.0000 








Table XXII 








Series II 














Comparison 














Stimulus 


q 


Q 


Q 1 


P 


p 


P 1 


84.80 


0.948 




0.0518 


0.005 


0.0052 




85.64 


0.932 




0.0641 


0.008 


0.0077 




86.48 


0.913 




0.0767 


0.013 


0.013 1 




87.32 


0.890 




0.0886 


0.016 


0.0160 




88.16 


0.863 




0.0986 


0.023 


0.0222 




89.00 


0.831 




0.1047 


0.032 


0.0301 




89.84 


0.796 


O.OOOI 


0.1054 


0.044 


0.0398 




90.68 


0.755 


0.0002 


0.1003 


0.059 


0.0513 




91-52 


0.712 


0.0007 


0.0893 


0.078 


0.0635 




92.36 


0.664 


0.0020 


0.0740 


O.IOI 


0.0761 




93-20 


0.614 


0.0048 


0.0565 


0.129 


0.0871 




94-04 


0.562 


O.OIOI 


0.0394 


0.161 


0.0949 




94.88 


0.509 


0.0187 


0.0248 


0.199 


0.0982 


O.OOOI 


95-72 


0.456 


0.0308 


0.0140 


0.242 


0.0953 


0.0004 


96.56 


0.403 


0.0456 


0.0070 


0.288 


0.0863 


0.0014 


97.40 


0.352 


0.0615 


0.0031 


0.339 


0.0723 


0.0039 


98.24 


0.304 


0.0764 


0.0012 


0.394 


0.0554 


0.0090 


99.08 


0.259 


0.0877 


0.0004 


0.449 


0.0384 


0.0182 


99.92 


0.218 


0.0944 


O.OOOI 


0.506 


0.0238 


0.0322 


100.76 


0.180 


0.0953 




0.563 


0.0131 


0.0507 


101.66 


0.147 


0.0914 




0.618 


0.0063 


0.0716 


102.44 


0.119 


0.0836 




0.672 


0.0026 


0.0915 


103.28 


0.094 


0.0732 




0.722 


0.0009 


0.1076 


104.12 


0.074 


0.0619 




0.768 


0.0003 


0.1 167 


10496 


0.057 


0.0505 




0.809 


O.OOOI 


0.1 186 


105.80 


0.043 


0.0399 




0.846 




0.1 132 


106.64 


0.032 


0.0309 




0.878 




0.1026 


107.48 


0.024 


0.0232 




0.904 




0.0886 


108.32 


0.017 


0.0170 




0.926 




0.0736 


2 




0.9999 


1. 0000 




1.0000 


0.9999 


R 




0.0000 


0.0000 




0.0000 


0.0000 



Table XXIII 



Series III 



Comparison 














Stimulus 


q 


Q 


Q 1 


p 


p 


p 1 


84.80 


0.941 




0.0592 


0.005 


0.0046 




86.06 


0.913 




0.0819 


0.008 


0.0084 




87.32 


0.876 


0.000 1 


0.1062 


0.015 


0.0148 




88.58 


0.830 


0.0007 


0.1279 


0.026 


0.0249 




89.84 


0.774 


0.0030 


0.1412 


0.042 


0.0396 




91.10 


0.709 


0.0095 


0.1408 


0.065 


0.0594 




92.36 


0.636 


0.0233 


0.1248 


0.099 


0.0836 


0.0001 


93-62 


0.557 


0.0461 


0.0965 


0.142 


0.1088 


0.0006 


94.88 


0.477 


0.0754 


0.0636 


0.197 


0.1294 


0.0029 


96.14 


0.397 


0.1043 


0.0349 


0.264 


0.1388 


0.0 1 00 


97.40 


0.322 


0.1245 


0.0156 


0.340 


0.1318 


0.0263 


98.66 


0.253 


0.1312 


0.0055 


0.423 


0.1083 


0.0543 


99.92 


0.193 


0.1239 


0.0015 


0.510 


0.0753 


0.0904 


101.18 


0.142 


0.1067 


0.0003 


0.596 


0.0431 


0.1251 


102.44 


0.102 


0.0848 


0.0001 


0.678 


0.0198 


0.1470 


10370 


0.070 


0.0630 




0.752 


0.0071 


0.1505 


104.96 


0.047 


0.0441 




0.816 


0.0019 


0.1367 


106.22 


0.030 


0.0293 




0.869 


0.0004 


O.I 122 


107.48 


0.019 


0.0186 




0.910 


0.0001 


O.0847 


108.74 


0.0 1 1 


0.01 14 




0.941 




O.O592 


2 




0.9999 


1.0000 




1. 0001 


1. 0000 


R 




0.0000 1 


0.0000 




0.0000 


O.OOOO 



Table XXIV 



Series IV 



Comparison 














Stimulus 


q 


Q 


Q 1 ' 


P 


P 


P 1 


84.80 


0.954 




0.0464 


0.008 


0.0076 




86.48- 


0.919 


0.0004 


0.0770 


0.015 


0.0151 




88.16 


0.868 


0.0026 


0.1157 


0.028 


0.0278 




89.84 


0.798 


0.0I2I 


0.1537 


0.051 


0.0480 




91.52 


0.710 ■ 


0.0370 


0.1762 


0.084 


0.0761 


0.0003 


93-20 


0.606 


O.0803 


0.1696 


0.133 


0.1096 


0.0019 


94.88 


0.495 


O.I297 


0.1320 


0.198 


0.1415 


0.0091 


96.56 


0.384 


O.1634 


0.079*3 


0279 


0.1603 


0.0293 


98.24 


0.282 


O.1672 


0.0357 


0.374 


0.1547 


0.0680 


99.92 


0195 


0.1434 


0.0113 


0.476 


0.1235 


0.1 194 


101.60 


0137 


0.1 166 


0.0024 


0.581 


0.0788 


0.1646 


103.28 


0.077 


0.0713 


0.0003 


0.680 


0.0387 


0.1851 


104.96 


0.044 


0.0424 




0.768 


0.0140 


0.1749 


106.64 


0.023 


0.0231 




0.840 


0.0036 


0.1433 


108.32 


O.OII 


0.0106 




0.896 


0.0006 


0.1041 


2 




1. 0001 


0.9999 




0.9999 


1. 0000 


R 




0.0000 


0.0000 




0.0001 


0.0000 



Table XXV 



Series V 



Comparison 














Stimulus 


q 


Q 


Q l 


p 


p 


P 1 


84.80 


0.947 


0.0006 


0.0526 


0.006 


0.0056 




86.90 


0.896 


0.0059 


0.0986 


0.015 


0.0149 




89.00 


0.815 


0.0288 


0.1570 


0.036 


0.0351 


0.0002 


91.10 


0.703 


0.0837 


0.2055 


0.076 


0.0716 


0.0025 


93-20 


0.568 


0.1566 


0.2101 


0.143 


0.1250 


0.0161 


95-30 


0.425 


0.2040 


0.1587 


0.242 


0.1813 


0.0586 


9740 


0.291 


0.1970 


0.0833 


0.370 


0.2099 


0.1315 


99-50 


0.181 


0.1493 


0.0280 


0.514 


0.1832 


0.1978 


101.60 


O.IOI 


0.0928 


0.0056 


0.656 


0.1 138 


0.2133 


103.70 


0.051 


0.0494 


0.0006 


0.779 


0.0465 


0.1757 


105.80 


0.023 


0.0227 




0.852 


O.OII2 


0.1380 


107.90 


0.009 


0.0092 




0.934 


O.O0I8 


0.0662 


2 




1. 0000 


1. 0000 




0.9999 


0.0999 


R 




0.0000 


0.0000 




O.OOOI 


0.0000 








Table XXVI 






Series VI 














Comparison 














Stimulus 


q 


Q 


Q 1 


p 


p 


P 1 


84.80 


0.939 


0.0024 


0.0610 


0.004 


0.0040 




87.32 


0.873 


0.0176 


0.1 193 


0.019 


0.0189 


O.OOOI 


89.84 


0.770 


0.0677 


0.1885 


0.039 


0.0381 


0.0013 


92.36 


0.632 


0.1509 


0.2323 


0.093 


0.0873 


0.013 1 


94-88 


0475 


0.2161 


0.2094 


0.189 


0.1610 


0.0619 


9740 


0.321 


0.2151 


0.1287 


0.329 


0.2272 


0.1556 


99-92 


0.193 


0.1602 


0.0490 


0.499 


0.2313 


0.2328 


102.44 


0.103 


0.0953 


0.0105 


0.669 


0.1553 


0.2299 


104.96 


0.048 


0.0466 


0.0012 


0.810 


0.0623 


0.1629 


107.48 


0.019 


0.0188 


O.OOOI 


0.906 


0.0132 


0.0890 


110.00 


0.007 


0.0070 




0.960 


0.0013 


0.0394 


112.52 


0.002 


0.0020 




0.986 


O.OOOI 


0.0140 


2 




0.9997 


1. 0000 




1.0000 


1.0000 


R 




0.0002 


0.0000 




0.0000 


0.0000 






Ti> 


BLE XXVII 








Series VII 














Comparison 














Stimulus 


q 


Q 


Q 1 


p 


p 


P 1 


84.80 


0.936 


0.0090 


O.0635 


0.006 


0.0062 




87.74 


0.851 


0.0546 


0.1395 


0.024 


0.0235 


O.OOI2 


90.68 


0.71 1 


0.1581 


0.2301 


0.070 


0.0684 


0.0158 


93-62 


0.529 


0.2494 


O.2672 


0.168 


0.1519 


0.0837 


96.56 


0.340 


0.2434 


O.I977 


0.328 


0.2456 


0.2067 


99.50 


0.185 


0.1623 


O.083I 


0.529 


0.2670 


0.2734 


102.44 


0.084 


0.0802 


O.OI73 


0.721 


0.1713 


0.2243 


105.38 


0.031 


0.0307 


O.OOI5 


0.865 


0.0572 


0.1260 


108.32 


0.009 


0.0094 


O.OOOI 


0.947 


0.0085 


0.0523 


1 1 1.26 


0.002 


0.0023 




0.983 


0.0005 


0.0166 


2 




0.9994 


1. 0000 




I.OOOI 


1. 0000 


R 




0.0006 


0.0000 




0.0000 


0.0000 



Table XXVIII 



Series J 'II I 



Comparison 














Stimulus 


q 


Q 


Q 1 


P 


P 


P 1 


84.80 


0.952 


0.0116 


0.0480 


0.005 


0.0048 


O.OOOI 


88.16 


0.865 


0.0775 


0.1288 


0.023 


0.0227 


0.0035 


91-52 


0.704 


0.2137 


0.2433 


0.080 


0.0779 


0.0409 


94.88 


0.491 


0.2927 


0.2952 


0.209 


0.1872 


0.1678 


98.24 


0.279 


0.2308 


0.2053 


0.416 


0.2943 


0.2979 


101.60 


0.126 


0.1 189 


0.0695 


0.648 


0.2678 


0.2768 


104.96 


0.043 


0.0427 


0.0096 


0.835 


0.1214 


0.1551 


108.32 


0.012 


0.0116 


0.0004 


0.942 


0.0225 


0.0580 


2 




0.9995 


1. 0001 




0.9986 


I.OOOI 


R 




0.0006 


0.0000 




0.0014 


0.0000 








Table XXIX 








Series IX 














Comparison 














Stimulus 


q 


Q 


Q l 


P 


P 


P 1 


84.80 


0.968 


0.0187 


0.0316 


0.004 


0.0044 


0.0002 


88.58 


0.876 


3.1358 


0.1203 


0.025 


0.0249 


0.0084 


92.36 


0.673 


0.3192 


0.2773 


0.097 


0.0940 


0.0801 


96.14 


0.397 


0.3120 


0.3443 


0.262 


0.2295 


0.2500 


99.92 


0.167 


0.1573 


0.1887 


0.507 


0.3280 


0.3295 


10370 


0.047 


0.0468 


0.0360 


0.752 


0.2400 


0.2203 


107.48 


0.009 


0.0086 


0.0018 


0.909 


0.0720 


0.0886 


1 1 1.26 


O.OOI 


O.OOIO 




0.977 


0.0070 


0.0229 


2 




0.9994 


1. 0000 




0.0998 


1. 0000 


R 




0.0006 


0.0000 




O.OOOI 


0.0000 








Table XXX 








Series X 














Comparison 














Stimulus 


q 


Q 


Q 1 


P 


P 


p 1 


84.80 


0.94s 


0.0544 


0.0550 


0.002 


0.0020 


0.0006 


89.00 


0.805 


0.2375 


0.1842 


0.021 


0.0214 


0.0287 


93.20 


0.548 


0.3583 


0.3435 


0.121 


0.1 183 


0.2130 


9740 


0.269 


0.2402 


0.3050 


o.377 


0.3239 


0.3999 


101.60 


0.088 


0.0860 


0.1023 


0.707 


0.3779 


0.2662 


105.80 


0.018 


0.0182 


0.0097 


0.919 


0.1440 


0.0796 


110.00 


0.002 


0.0024 


0.0002 


0.988 


0.0125 


0.0120 


2 




0.0970 


0.9999 




1. 0000 


1. 0000 


R 




0.0031 


0.0000 




O.OOOI 


0.0000 



Table XXXI 



subject I; one table being given to each of the ten series [I-X]. 
In the first columns are found the intensities of the comparison 
stimuli used in each particular series. The second columns give 
the probabilities of a lighter judgment on these comparison 
stimuli. The third and fourth columns give, under the head- 
ings Q and Q 1; the probabilities with which the different com- 



SAMUEL W. FERNBERGER 



parison stimuli appear as determinations of the just percep- 
tible and of the just imperceptible negative difference. The next 
column gives the probabilities with which heavier judgments 
may be expected on the stimuli. These last values serve as 
a basis for the calculation of the probabilities of the different 
stimuli for being observed as determinations of the just per- 
ceptible and of the just imperceptible positive difference, the 
values of which are found in the last columns of the table. 
The numbers at the bottom of each column — marked 2 — give the 
sums of all of the terms and the numbers R give the values 
of the remainder. These remainders and the sums of the proba- 
bilities when added up come to unity within the limit of o.oooi 
and thus prove the correctness of the computation. Tables 
XXXII-XLI are similarly constructed and give the correspond- 
ing values of subject II. 

It will be noticed that the size of the remainder is in no case so 
large as to invalidate the series. There is a tendency for the 
size of the remainder to increase as the series decreases in length. 
A glance at the table that follows, which contains all the remain- 
ders for both subjects, shows this tendency. In the first four 



Series I (G) 












Comparison 














Stimulus 


q 


Q 


Q 1 


p 


p 


P 1 


84.80 


0.921 


0.0007 


0.0786 


0.0 10 


0.0100 




8774 


0.822 


0.0036 


0.1642 


0.031 


0.0305 




90.26 


0.692 


0.0099 


0.2331 


0.074 


0.0712 




92.36 


0.560 


0.0183 


0.2308 


0.136 


0.12 10 




94.04 


0.448 


0.0265 


0.1618 


0.207 


0.1587 


O.OOOI 


95-30 


0.367 


0.0342 


0.0833 


0.272 


0.1653 


O.OOOI 


96.14 


0.315 


0.0429 


0.0330 


0.320 


0.1418 


0.0006 


96.56 


0.291 


0.0558 


0.0108 


0.346 


0.1042 


0.0016 


96.98 


0.267 


0.0699 


0.0032 


0.371 


0.0733 


0.0040 


9740 


0.244 


0.0847 


0.0009 


0.398 


0.0494 


0.0097 


97.82 


0.223 


0.0995 


0.0002 


0.425 


0.0317 


0.0218 


98.24 


0.202 


0.1 132 


0.0001 


0.453 


0.0195 


0.0457 


98.66 


0.183 


0.1254 




0.481 


0.01 13 


0.0902 


99-50 


0.148 


0.1 192 




0.532 


0.0065 


0.1 501 


100.76 


0.105 


0.0941 




0.619 


0.0035 


0.1095 


102.44 


0.062 


0.0596 




0.720 


0.0015 


0.2036 


104.54 


0.030 


0.0292 




0.824 


0.0005 


0.1550 


107.06 


0.0 10 


0.0105 




0.912 


O.OOOI 


0.0853 


110.00 


0.003 


0.0026 




0.067 




0.0328 


2 




0.9998 


1.0000 




1. 0000 


1. 0001 


R 




0.0001 


0.0000 




0.0000 


0.0000 



Table XXXII 



Series II 



Comparison 














Stimulus 


q 


Q 


Q 1 


p 


P 


P 1 


84.80 


0.938 




0.0621 


0.013 


0.0130 




85.64 


0.919 




0.0763 


0.018 


0.0178 




86.48 


0.897 




0.0890 


0.025 


0.0240 




87.32 


0.871 




0.0998 


0.033 


0.0316 




88.16 


0.841 




0.1072 


0.045 


0.0408 




89.00 


0.806 


O.OOOI 


0.1098 


0.058 


0.0510 




89.84 


0.767 


O.OOOI 


0.1061 


0.076 


0.0621 




90.68 


0.725 


0.0005 


0.0963 


0.096 


0.0731 




91-52 


0.679 


0.0015 


0.0814 


O.I2I 


0.0829 




92.36 


0.629 


0.0037 


0.0638 


O.I49 


0.0901 




93-20 


0.578 


0.0080 


0.0457 


O.182 


0.0937 




94.04 


0.525 


0.0153 


0.0297 


0.219 


0.0921 


O.OOOI 


94-88 


0.472 


0.0261 


0.0173 


O.260 


0.0854 


0.0004 


9572 


0.420 


0.0400 


0.0090 


O.305 


0.0739 


0.0012 


96.56 


0.368 


0.0555 


0.0041 


0.353 


0.0595 


0.0031 


97.40 


0.319 


0.0708 


0.0016 


0.403 


0.0439 


0.0071 


08.24 


0.273 


0.0834 


0.0006 


0.455 


0.0296 


0.0141 


99.08 


0.231 


0.0916 


0.0002 


0.508 


0.0180 


0.0252 


99.92 


0.193 


0.0946 


O.OOOI 


0.559 


0.0098 


0.0404 


100.76 


0.158 


0.0922 




0.612 


0.0047 


0.0581 


101.60 


0.128 


0.0856 




0.661 


0.0020 


0.0766 


102.44 


0.102 


0.0762 




0.708 


0.0007 


0.0931 


103.28 


0.081 


0.0653 




0.752 , 


0.0002 


0.1 054 


104.12 


0.062 


0.0539 




0.796 


O.OOOI 


0.1089 


104.96 


0.048 


0.0433 




0.828 




0.III2 


105.80 


0.036 


0.0337 




0859 




0.1055 


106.64 


0.027 


0.0256 




0.887 




0.0959 


107.48 


0.019 


0.0191 




0.910 




0.0835 


108.32 


0.014 


0.0139 




0.930 




0.0703 


2 




1. 0000 


1. 0001 




1. 0000 


1. 0001 


R 




0.0000 


0.0000 




0.0000 


0.0000 






1 


ABLE XXXI 


[I 






Series III 














Comparison 














Stimulus 


q 


Q 


Q 1 


P 


p 


P 1 


84.80 


0.933 




0.0673 


0.016 


0.0162 




86.06 


0.902 




0.0910 


0.026 


0.0260 




87.32 


0863 


0.0002 


0.1155 


0.041 


0.0395 




88.58 


0.814 


O.OOII 


0.1351 


0.062 


0.0573 




89.84 


0.755 


0.0042 


0.1446 


0.091 


0.0785 




91.10 


0.688 


0.0123 


0.1394 


0.129 


0.1006 


0.0002 


92.36 


0.614 


0.0284 


0.1 187 


0.176 


0.1200 


0.0009 


93.62 


0.535 . 


0.0533 


0.0876 


0.233 


0.1308 


0.0036 


04.88 


0.454 


0.0829 


0.0550 


0.298 


0.1286 


0.0109 


96.14 


0.376 


O.I 100 


0.0286 


0.372 


0.1 125 


0.0262 


9740 


0.303 


0.1270 


0.0120 


0.450 


0.0855 


0.0509 


08.66 


0.236 


0.1297 


0.0040 


0.530 


' 0.0554 


0.0820 


99.92 


0.179 


0.1 198 


0.0010 


0.609 


0.0299 


O.I 120 


101.18 


0.13 1 


O.IOI2 


0.0002 


0.684 


0.013 1 


0.1326 


102.44 


0.093 


0.0790 




o.75i 


0.0046 


0.1387 


103.70 


0.064 


0.0579 




0.81 1 


0.0012 


0.1302 


104.96 


0.042 


0.0402 




0.860 


0.0002 


0.1117 


106.22 


0.027 


0.0263 




0.000 


O.OOOI 


0.0887 


107.48 


0.018 


0.0165 




0.931 




0.0656 


108.74 


O.OIO 


0.0099 




0.954 




0.0459 


2 




0.0999 


1. 0000 




1. 0000 


I.OOOI 


R 




o.oooo 


0.0000 




0.0000 


0.0000 



Table XXXIV 



Series IV 



Comparison 














Stimulus 


q 


Q 


Q 1 


p 


p 


P 1 


84.80 


0.922 


0.0002 


0.0780 


0.018 


0.0183 




86.48 


0.874 


0.0014 


0.1 149 


0.034 


0.0330 




88.16 


0.812 


0.0069 


0.1514 


0.058 


0.0552 




89.84 


0.733 


0.0232 


0.1753 


0.095 


0.0849 


0.0004 


91-52 


0.639 


0.0560 


0.1736 


0.147 


0.1 186 


0.0025 


93-20 


0.536 


O.IOII 


0.1425 


0.215 


0.148 1 


0.0106 


94.88 


0.430 


0.1423 


0.0937 


0.297 


0.1612 


0.0318 


96.56 


0.329 


0.1623 


0.0474 


0.393 


0.1496 


0.0698 


98.24 


0.239 


0.1552 


0.0177 


0.496 


0.1 145 


0.1171 


99.92 


0.165 


0.1282 


0.0046 


0.598 


0.0697 


0.1561 


101.60 


0.107 


0.0935 


0.0008 


0.694 


0.0326 


0.1710 


103.28 


0.066 


0.0615 


O.OOOI 


0.779 


O.OI 12 


0.1588 


104.96 


0.038 


0.0370 




0.848 


0.0027 


0.1290 


106.64 


0.021 


0.0206 




0.902 


0.0004 


0.0920 


108.32 


O.OII 


0.0106 




0.939 


O.OOOI 


0.0609 


2 




1. 0000 


1.0000 




1. 0001 


1. 0000 


R 




0.0000 


0.0000 




0.0000 


0.0000 



Table XXXV 



Series V 














Comparison 














Stimulus 


q 


Q 


Q 1 


P 


P 


P 1 


84.80 


0.906 


0.0022 


0.0944 


O.OO9 


0.009I 




86.90 


0.838 


0.0125 


0.1465 


0.022 


0.0220 




89.00 


0.746 


0.0437 


0.1932 


O.O49 


0.0472 


0.0004 


91.10 


0.631 


O.IOOI 


0.2090 


O.96 


O.0887 


0.0041 


93.20 


0.503 


0.1605 


0.1775 


O.I7I 


O.I42I 


0.0219 


95.30 


0.375 


0.1912 


O.I 122 


0.274 


O.I895 


0.0697 


97.40 


0.259 


0.1784 


O.O498 


O.403 


O.2O20 


0.1424 


99-SO 


0.165 


0.1364 


O.OI45 


0.542 


O.1624 


0.2013 


101.60 


0.097 


0.0884 


O.OO26 


O.677 


O.O928 


0.2102 


103.70 


0.052 


0.0501 


0.0OO3 


O.792 


0.035I 


0.1711 


105.80 


0.025 


0.0251 




O.878 


O.O080 


0.1 143 


107.90 


O.OII 


0.0113 




0.935 


O.OOII 


0.0646 


2 




0.9999 


I. OOOO 




1. 0000 


1. 0000 


R 




0.0002 


O.OOOO 




O.OOOI 


0.0000 



Table XXXVI 



Series VI 



Comparison 
Stimulus 


q 


Q 


Q 1 


p 


p 


P l 


84.80 

87.32 

89.84 

92.36 

94.88 

97.40 

99.92 

102.44 

104.96 

107.48 

110.00 

112.52 


0.890 
0.806 
0.692 
0.556 
0.413 
0.280 

0.173 
0.096 
0.048 
0.021 
0.008 
0.003 


0.0069 
0.0321 
0.0895 
0.1621 
0.2047 
0.1929 
0.1440 
0.0884 
0.0462 
0.021 1 
0.0084 
0.0030 


O.IIOI 

0.1725 

0.2208 
0.2204 
0.1622 
0.0821 
0.0264 
0.0050 
0.0005 


0.025 
0.055 

0.109 

0.193 
0.307 
0.444 
0.588 

0.721 
0.829 
0.906 

0.953 
0.979 


0.0250 
0.0540 
0.1004 
0.1580 
0.2033 
0.2038 
0.1502 

0.0759 

0.0244 
0.0046 
0.0004 


0.0008 
0.0069 
0.0326 
0.0914 

0.1652 
0.2083 
0.1956 
0.1446 

0.0881 
0.0458 
0.0206 


2 
R 




0.9993 
0.0008 


1. 0000 
0.0000 




1. 0000 
0.0000 


0.9999 

0.0000 



Table XXXVII 



Series I'll 



Comparison 
Stimulus 


q 


Q 


Q 1 


p 


P 


P 1 


84.80 

8774 

90.68 

93.62 

96.56 

99.50 

102.44 

105.38 

108.32 

111.26 


0.954 
0.873 
0.726 

0.523 
0.313 
0.151 
0.058 
0.017 
0.004 
0.00 1 


0.0085 
0.0615 
0.1861 
0.2807 
0.2451 
0.1395 
0.0566 
0.0171 
0.0039 
0.0007 


0.0459 
0.1208 
0.2287 
0.2886 
0.2170 
0.0840 
0.0141 
0.0009 


0.009 
0.029 
0.082 
0.164 
0.343 
0.536 
0.720 
0.860 
0.942 
0.081 


0.0086 
0.0292 
0.0785 
0.1451 
0.2534 
0.2601 
0.1622 
0.0541 
0.0083 
0.0005 


0.0014 
0.0159 
0.0879 
0.2015 
0.2656 
0.2221 
0.1297 
0.0567 
0.0192 


2 
R 




0.9997 
0.0004 


1. 0000 
0.0000 




1. 0000 
0.0000 


1. 0000 
0.0000 



Table XXXVIII 



Series VIII 














Comparison 














Stimulus 


q 


Q 


Q 1 


p 


p 


P 1 


84.80 


0.938 


0.0166 


0.0621 


O.OII 


0.0114 


0.0003 


88.16 


0.839 


0.0923 


0.1508 


0.041 


0.0406 


0.0065 


91-52 


0.672 


0.2252 


0.2582 


0.115 


0.1092 


0.0517 


94-88 


0.460 


0.2855 


0.2856 


0.254 


0.2130 


0.1716 


98.24 


0.259 


0.2169 


0.1803 


0.450 


0.2819 


0.2806 


101.60 


0.117 


0.1 105 


0.0557 


0.661 


0.2272 


0.2623 


104.96 


0.041 


0.0406 


0.0070 


0.829 


0.0968 


0.1590 


108.32 


O.OII 


O.OII2 


0.0003 


0.932 


0.0185 


0.0680 


2 




O.9988 


1. 0000 




0.9986 


1. 0000 


R 




O.OOII 


0.0000 




0.0014 


0.0000 



Table XXXIX 



Series IX 



Comparison 














Stimulus 


q 


Q 


Q l 


p 


p 


P 1 


84.80 


0.952 


0.0287 


0.0478 


0.004 


0.0040 


0.0006 


88.58 


0.840 


0.1586 


0.1521 


0.028 


0.0281 


0.0200 


92.36 


0.628 


0.3179 


0.2980 


0.121 


0.1 175 


0*488 


96.14 


0.36s 


0.2912 


0.3188 


0.334 


0.2837 


0.3383 


99.92 


0.155 


0.1463 


0.1549 


0.622 


0.3523 


0.3091 


103.70 


0.046 


0.0454 


0.0271 


0.853 


0.1829 


0.1408 


107.48 


0.009 


0.0092 


0.0013 


0.963 


0.0304 


0.0367 


1 1 1.26 


O.OOI 


O.OOI2 




0.994 


O.OOII 


0.0058 


2 




0.9985 


1. 0000 




I. OOOO 


1. 000 1 


R 




0.0014 


0.0000 




0.000 1 


0.0000 




Table XL 



12 



SAMUEL W. FERNBERGER 



Series X 



Comparison 
Stimulus 


q 


Q 


Q 1 


p 


p 


P 1 


84.80 

89.00 

93-20 

97.40 

101.60 

105.80 

110.00 


0.952 
0.828 
0.592 
0.315 
0.116 
0.028 
0.004 


0.0300 
0.1978 
0.3468 
0.2697 
0.1 123 
0.0280 
0.0044 


0.0483 
0.1634 
0.3215 
0.3196 
0.1 301 
0.0166 
0.0005 


0.014 
0.061 

0.183 

0.400 

0.653 
0.851 
0.954 


0.0140 
0.0598 
0.1699 
0.3022 
0.2967 
0.1340 
0.0224 


0.0023 
0.0366 

0.1733 
0.3188 

0.2817 
0.1418 

0.0455 


2 
R 




0.9980 
0.0020 


1. 0000 
0.0000 




0.9990 

O.OOIO 


1. 0000 
0.0000 



Table XLI 

series there are only two remainders and these are only 0.0001. 
In the later and shorter series the remainders are very much 
larger and occur much more frequently. It will also be noticed 
that there are no remainders for either subject for the just im- 
perceptible differences both positive and negative. This of course 
indicates that our series were in every case extended enough so that 
these imperceptible differences would always fall upon one of 
the stimuli of our series, within the limit of 0.0001. It would also 
seem that the remainders for the just perceptible negative differ- 
ence are on the whole greater than those of the just perceptible 
positive difference. 





Subj 


ect I 


Subject II 




Just 


Just 


Just 


Just 




perceptible 


perceptible 


perceptible 


perceptible 




negative 


positive 


negative 


positive 


Series 


difference 


difference 


difference 


difference 


I 


0.0000 


0.0000 


O.OOOI 


0.0000 


IV 


0.0000 


O.OOOI 


0.0000 


0.0000 


V 


0.0000 


O.OOOI 


0.0002 


O.OOOI 


VI 


0.0002 


0.0000 


0.0008 


0.0000 


VII 


0.0006 


0.0000 


0.0004 


0.0000 


VIII 


0.0006 


0.0014 


O.OOII 


0.0014 


IX 


0.0006 


O.OOOI 


0.0014 


O.OOOI 


X 


0.0031 


O.OOOI 


0.0020 


O.OOIO 



When we have once obtained the probabilities for the different 
comparison stimuli with which these four differences may occur, 
we may derive the most probable value of each of the just percepti- 
ble and just imperceptible differences very easily. Multiplying 
each value of the comparison stimulus by its probability and adding 



METHOD OF JUST PERCEPTIBLE DIFFERENCES 73 

these products we obtain [xQ], [xQJ, [xP], and [xPJ. These 
are respectively the most probable values of the just perceptible 
negative difference, the just imperceptible negative difference, 
the just perceptible positive difference and the just imperceptible 
positive difference, calculated on the basis of our experiments by 
the method of constant stimuli. Tables XLII and XLIII give 
these values of the four differences for subjects I and II. Op- 
posite the numbers for the series, which are found in the first 
columns, we have in the 3d, 5th, 7th, and 9th columns the calcu- 
lated values for these differences. 

The observed values of these fundamental differences are found 
in the other columns of these tables. The observed and calculated 
values for each series and for every difference, are found directly 
next to one another so that they may be more easily compared. 
These observed values are calculated very readily and in the man- 
ner explained in our description of the experiment, given above. 
The values for the just perceptible and imperceptible negative and 
positive differences are ascertained for each group in every series, 
by picking out each value according to the definitions of these 
quantities. There were ten groups in each series, five ascending 
and five descending, so that we will obtain in each case, ten values 
for each difference. These ten values are averaged and the result 
is given as the value of the difference. The ease with which the 
differences are calculated after we have our experimental data 
is one of the great advantages of the method of just perceptible 
differences over the other methods of psychophysical measure- 
ment. In this method, the measure of sensitivity is obtained by 
the simple solving of four averages, which can be performed in 
a very short space of time. With the method of constant stimuli, 
on the other hand, a long and complicated series of calculations 
must be entered into before these same values are obtained. 

We now calculate the measure of accuracy that is shown by our 
observed results of the method of just perceptible differences by 
means of the probable error. If our threshold is found to be 95 
grams and the probable error ± 0.50 grams, it indicates that this 
threshold will fall within the interval of 94.50 grams and 95.50 
grams with the probability of 0.50. This measure of precision is 



2 8 .3 



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METHOD OF JUST PERCEPTIBLE DIFFERENCES 



75 



calculated only for the thresholds and is obtained by the formula 

It? 



P. E. 



0.6745 3 



n(n-i) 

In this equation 2v 2 is the sum of the variations of the different 
thresholds from the average, and n is the number of thresholds 
considered. Table XLIV gives the probable errors for both sub- 





Subj 


ect I 


Subject II 




Threshold in Threshold in 


Threshold in 


Threshold in 




direction of 


direction of 


direction of 


direction of 


Series 


decrease 


increase 


decrease 


increase 


I 


±0.50 


±0.43 


±0.43 


±0.44 


II 


±0.88 


±0.82 


±0.87 


±o.75 


III 


±0.78 


±0.83 


±0.79 


±0.83 


IV 


±0.96 


±0.72 


±0.83 


±0.61 


V 


±0.72 


±0.66 


±0.65 


±0.65 


VI 


±0.27 


±0.64 


±0.62 


±0.52 


VII 


±0.53 


±0.65 


±0.50 


±0.63 


VIII 


±0.52 


±0.56 


±0.57 


±0.56 


IX 


±0.56 


±0.60 


±0.62 


±0.76 


X 


±0.55 


±0.54 


±0.60 


±0.72 



Table XLIV 

jects for the two thresholds of increase and decrease. It will be 
noticed that these values are comparatively constant. Considering 
the size of the values, the precision of which they are measuring, 
these quantities are small. In no case is the probable error greater 
than one per cent of the threshold and very often it is close to 
one-half a per cent or even less. Thus we conclude that the thres- 
holds in the method of just perceptible differences fall with a high 
probability within a comparatively limited space. 

The sums of the products xP, xP 1 , xQ and xQ 1 give the most 
probable values of the just perceptible and just imperceptible posi- 
tive and negative differences, but a finite number of observations 
can not be expected to give exactly these results. The outcome of 
a series depends on chance influences, which prevent us from 
obtaining the calculated result exactly, and we must be satisfied 
with determining the limit inside of which we may expect the 
results to fall with a given probability. The solution of this prob- 
lem is given by the theorem of Tchebicheff, which applies 
to problems of this kind (cfr. Urban, Statistical Methods, pp. 65, 
and Archiv f. d. ges. Psychologie, Vol. 15, pp. 302-304). We 



76 SAMUEL W. FERNBERGER 

give the formulae for the just perceptible and the just impercep- 
tible positive difference only, because their form for the two other 
differences is perfectly obvious. Suppose we make n determina- 
tions of each one of these quantities with a certain series of com- 
parison stimuli. We then may expect with a probability exceed- 
ing i — — that the arithmetical mean of all the observations of 
the just perceptible difference will be within the limits 

%xP ± t V2* 2 />— {%xP) 2 

and that of the just imperceptible positive difference between the 
limits 

SxP 1 ± -i/lx^P 1 — CZxPy . 

Applying the same theorem to the determination of the upper limit 
of the interval of uncertainty we find that there exists a proba- 
bility exceeding i — £. that the actually observed result will be 
found between the limits 

The difference 2x 2 P — (2xP) 2 plays a part similar in the 
theorem of Tchebicheff to that of the product 2spq in the 
theorem of Bernoulli, since the square roots of both these 
terms determine the size of the interval inside of which the 
result may be expected with a given probability. The actual 
determination of these intervals requires the square root of these 
differences, but this problem is of less interest than the question 
as to the comparative size of these intervals in series of different 
length. For this purpose it is sufficient to give these differences as 
it is done in Table XLV for Subject I and in Table XL VI for 
Subject II. These tables also contain the values of the sums of 
the terms xxP which is needed for the calculation of the differ- 
ences in question. These quantities also enable us to determine 
the limits for the result of a series of determinations of the upper 

* The theorem of Tchebicheff enables us merely to determine the limits 
inside of which we may expect the result of an actual observation to fall with 
a given probabilicy. Wirth [Zur erkenntnisstheoretischen und mathema- 
tischen Begrilndung der Massmethoden fur die U nterschiedsschwelle , Archiv 
f. d. ges. Psychologie, 191 1, Vol. 20, p. 84] seems to believe that this theorem 
might enable us to find relations between different values of the psycho- 
metric functions, but there is no possibility of doing so. 



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78 



SAMUEL W. FERNBERGER 



limit of the interval of uncertainty. In our experiments we have 
in all the cases n = 5, so that the radicals for the limits of the 
interval of uncertainty can be found by averaging the corres- 
ponding values in the tables. 

Table XL VII gives these values for both subjects. The greater 





Subject I 


Subject II 


Series 


d 1 


d 11 


d 1 


d" 


I 

II 

Ill 

IV 

V 

VI 

VII 

VIII 

IX 

X 


8.42 
10.56 
12.26 
13,26 
I4-30 
18.08 
18.42 
17.00 
17.14 
21.70 


8.30 

9.02 

12.36 

13-47 
12.74 
18.92 
17.82 
12.85 
20.70 
10.66 


6.85 
9.84 
11.48 
13.76 
I5-IO 
16.91 
1378 
14-54 
12.80 
19.04 


8.22 
6.68 
1366 
14-74 
14.58 
20.36 
19.18 
1370 
14.82 
21.21 



Table XLVII 

the value, the greater will be the interval within which we may 
expect our determination of the threshold to fall with a given 
probability and therefore the less accurate that determination 
will be. These quantities definitely tend to increase in size as the 
series become shorter, and therefore, the determinations of our 
thresholds by short series of experiments are not so accurate as 
those obtained by more extended series. Furthermore, it would 
definitely appear that the determinations of our thresholds by the 
graded series I are more accurate than any of the others. 

We are now in the possession of all the data necessary for the 
comparison of the two methods under discussion. The thres- 
holds are the important values in these methods, since they are 
the basis for the determination of the sensitivity of the subject. 
It is by the means of the values of the thresholds, then, that we 
will compare the methods of just perceptible differences and of 
constant stimuli. Tables XL VIII and XLIX show the values 
of the thresholds for both subjects, obtained in the different ways 
that have been described above. The first columns in these tables 
give the numbers of the series into which the results of the method 
of just perceptible differences were divided. The values of the 
threshold in the direction of decrease are found in the next three 
columns. The first of these are the values obtained from the re- 



Subject I 





Threshold in the direction of decrease 


Threshold in the direction of increase 




Method of 
constant 
stimuli 


Method of Just Percep- 
tible Differences 


Method of 
constant 
stimuli 


Method of Just Percep- 
tible Differences 


Series 


Calculated 


Observed 


Calculated 


Observed 


I 

II 

Ill 

IV 

V 

VI 

VII 

VIII 

IX 

X 


95.02 
95.02 
94-52 
94-8 1 
94.20 
94.48 
94.06 
94-74 
94-75 
93-89 


95-96 
95-33 
94-85 
94.99 

94-37 
94-59 
94.24 
0483 
94-99 
94.10 


96.81 
95.26 
93.87 
97-40 
93-62 
93-02 
96.71 
94-88 
96.14 
94.88 


100.10 
99.82 
99-77 

100.30 
99.30 
99-93 
99.09 
99-45 
99.82 

98.93 


99.07 
99.26 
99-45 
99.76 
99-13 
99-88 
99.07 
99-33 
99-78 
98.93 


98.58 

99-58 

98.28 

101.26 

97.92 

99.04 

102.58 

100.26 

101.81 

101.18 



Table XLVIII 



Subject II 





Threshold in the direction of decrease 


Threshold in 


the direction of increase 




Method of 
constant 
stimuli 


Method of Just Percep- 
tible Differences 


Method of 

constant 

stimuli 


Method of Just Percep- 
tible Differences 


Series 


Calculated 


Observed 


Calculated 


Observed 


I 

II 

Ill 

IV 

V 

VI 

VII 

VIII 

IX 

X 


93-26 
94-44 
94.16 
9376 
93-24 
93-34 
93-93 
94.27 
94.20 
94-57 


94.81 
94.89 
94-55 
94-13 
93-63 
93-84 
94.02 
94.40 
94-28 
94-57 


95-74 
94.42 

94-63 
04.88 
94.04 
95-88 
95-53 
95-88 
93-30 
96.56 


98.94 
98.96 
98.19 
98.32 
98.86 
98.37 
98.96 
99.00 
98.34 
99.04 


98.34 
97-94 
97-98 
98.07 
98.64 
97-44 
98.99 
98.85 
98.18 
98.97 


97-25 
98.74 
98.03 
98.24 
97-50 
100.05 

99-79 

100.09 

99.92 

98.87 



Table XLIX 
suits of the method of constant stimuli, that were taken simul- 
taneously with the different series of the method of just percepti- 
ble differences. In the next column are the calculated thresholds 
of the method of just perceptible differences, and in the third 
column are the observed values for the same method. The second 
halves of the tables show the same arrangement of values for the 
threshold in the direction of increase. 

An examination of these tables shows unsystematic variations 
between the different results that are to be expected. That is, 
within a given set, some individual results are greater and some are 
smaller than the average, but these variations occur in a haphazard 
manner and not in a regular way. The larger the amount of data 
that goes to determine any value or set of values, the smaller these 
chance variations should become. This is borne out by our re- 



80 SAMUEL W. FERNBERGER 

suits, as the random variations are greatest in the observed results 
of the method of just perceptible differences, which values are 
determined by smaller experimental data than are the others. But 
these variations are comparatively small and seem to point to a 
high degree of similarity between the two methods. 

There is one observation, however, that may be of significance. 
If we compare the observed thresholds of the method of constant 
stimuli and the calculated thresholds of the method of just per- 
ceptible differences, we find that the variations, in the case of 
both subjects, are always in one direction. The calculated thres- 
holds in the direction of decrease by the method of just percepti- 
ble differences are constantly larger than the corresponding values 
in the method of constant stimuli; and for the threshold in the 
direction of increase, they are constantly smaller. Thus the limits 
of the interval of uncertainty are constantly narrowed in the just 
perceptible difference method when compared with the same in- 
terval of the other method. This is a serious fault as it is just this 
interval that is the measure of sensitivity of our subject. If the 
values of one method had been constantly greater than those 
of the other method for both thresholds, the interval of uncer- 
tainty would have remained the same although the point of sub- 
jective equality would have changed. It will be noticed that these 
systematic variations are greatest in the long series of short steps, 
while in the short series of large steps they become so small that 
they can be practically disregarded. The variation is greatest in 
series I, which was performed in an effort to test out a graded 
approach of the central values of the series. This may be due 
to the fact that by a carefully graded approach, we emphasize be- 
fore hand the probability that the differences will fall on the 
central values. The greater the number of results that go to make 
up a value, the more nearly should that value coincide with a 
calculated probability. If this be true, then the values of the 
series I to IV should be more nearly correct than the series VIII 
to X. It is a curious fact that the more nearly the experimental 
arrangement of the method of just perceptible differences ap- 
proaches that of the method of constant stimuli, the closer do the 
values under discussion coincide. The experimental arrangement 



METHOD OF JUST PERCEPTIBLE DIFFERENCES 81 

of the method of constant stimuli consists of a small number of 
pairs of stimuli with comparative large differences of intensity in 
the steps of the series. This is like the arrangement of series X, 
and here the values of the thresholds for the two methods prac- 
tically agree. 

Series II has quite a different arrangement ; having many steps 
of a small interval and here the greatest discrepancies are found 
between the two methods. Thus there seem to be discrepancies that 
may be due, either to the length of the series, or to the attitude 
with which the subject approached the two methods. Our form 
of experimentation tried to eliminate any difference in attitude and 
was probably almost entirely, but not absolutely, successful. If we 
had been able to devise a means by which the attitude of the sub- 
ject was identical for both methods, it seems very probable that 
our results would have showed a still closer agreement. 

In practically all of the studies by these methods, it has been 
found that a higher sensitivity was obtained by the method of 
just perceptible differences than by the method of constant stimuli. 
If the difference in our values under discussion are significant, we 
could then account for the discrepancies in the results of the 
former studies, as a fundamental difference in the methods 
themselves. 

But these differences are not large enough to have us disregard 
the method of just perceptible differences as a practical method 
of psychophysical measurement. It seems wisest, however, to 
disregard the form of a graded approach of the central values, 
as the method in this form seems to show its greatest discrepan- 
cies. It would also seem that a short series of large steps is pre- 
ferable to a long series of small steps. The best form of experi- 
mentation with this method would be to take a considerable num- 
ber of series, each series to consist of not more than ten steps and 
with a considerable difference of weight between the steps. If 
such a series is used, the results of this study seem to indicate, that 
any discrepancies between the results obtained in this way and 
those of the method of constant stimuli taken under identical con- 
ditions would be so small that they could be disregarded. 



